Introducing a custom Reinforcement Learning - Sequential Sampling Model (RLSSM) into HSSM¶

Before proceeding with this tutorial, we recommend going through the main RLSSM tutorial (Tutorial for hierarchical Bayesian inference for Reinforcement Learning - Sequential Sampling Models for a general familiarity with RLSSM modeling in HSSM.

This tutorial demonstrates how to incorporate custom, user-defined Reinforcement Learning Sequential Sampling Models (RLSSM) into the HSSM framework by modifying the rldm.py file. The tutorial walks through the key steps needed to define, implement, and integrate your custom RLSSM likelihood functions.

More specifically, this tutorial shows how to add a custom model on a 2-armed bandit environment. Our model employs simple Rescorla-Wagner-style learning updates with two learning rates - for positive and negative prediction errors. The decision process is a drift-diffusion model with collapsing bounds ('angle' model).

⚠️ Warning¶

You will have to edit rldm.py file and re-install hssm for this tutorial to work. Running this notebook without changes to the rldm.py file will throw errors.

In [15]:

# Import necessary libraries
import numpy as np
import arviz as az
import matplotlib.pyplot as plt
from functools import partial

# Import HSSM and simulator package
import hssm
from hssm.utils import decorate_atomic_simulator
from hssm.likelihoods.rldm import make_rldm_logp_op
from hssm.distribution_utils.dist import make_hssm_rv
from ssms.basic_simulators.simulator import simulator

# Import necessary libraries import numpy as np import arviz as az import matplotlib.pyplot as plt from functools import partial # Import HSSM and simulator package import hssm from hssm.utils import decorate_atomic_simulator from hssm.likelihoods.rldm import make_rldm_logp_op from hssm.distribution_utils.dist import make_hssm_rv from ssms.basic_simulators.simulator import simulator

In [2]:

# Set the style for the plots
plt.style.use('seaborn-v0_8-dark-palette')

# Set the style for the plots plt.style.use('seaborn-v0_8-dark-palette')

Section I: Introduce the likelihood function for your custom model in HSSM¶

In the existing implementation, rldm.py already defines the model config (Step 1) and the likelihood function (Step 3) for the model mentioned above. The model is called rlssm2 in the existing implementation. An easy way to follow this tutorial is simply considering rlssm2 as your custom model that you want to incoporate into HSSM. Make the necessary changes outlined in steps 2, 4 and 5 to readily use the implemented model.

Step 1: Define Your Custom RLSSM Model Configuration¶

Location: Add to the rlssm_model_config_list dictionary

Purpose: Define the meta-data and parameters of your custom model in a configuration dictionary.

Details:

• Create a new entry in the rlssm_model_config_list dictionary with a unique model name
• Specify the following required fields:
  • name: Name your custom rlssm model
  • description: Optional description of the model
  • n_params: Number of model parameters
  • n_extra_fields: Number of extra_fields columns passed to the likelihood function (e.g., trial, feedback)
  • list_params: List of parameter names in the order they'll be passed to the likelihood function
  • extra_fields: List of extra_fields columns
  • decision_model: Type of the likelihood for the decision process model (typically "LAN" - likelihood approximation networks)
  • LAN: Specific LAN model to use (e.g., "angle")

Example:

"my_custom_rlssm": {
    "name": "my_custom_rlssm", 
    "description": "Custom RLSSM with special features", 
    "n_params": 8, 
    "n_extra_fields": 3, 
    "list_params": ["param1", "param2", "param3", ...], 
    "extra_fields": ["extra_fields1", "extra_fields2", "extra_fields3", ...], 
    "decision_model": "LAN", 
    "LAN": "angle", 
}

Step 2: Specify Which Model Configuration to Use¶

Location: Update the MODEL_NAME variable

Purpose: Inform the HSSM package which model configuration to use from your defined list.

Details:

• Set MODEL_NAME to match one of the keys in your rlssm_model_config_list dictionary

• The system will automatically load the corresponding configuration

• This makes it easy to switch between different model variants

Example:

MODEL_NAME = "my_custom_rlssm"  # Must match a key in rlssm_model_config_list

Step 3: Implement Your Custom Likelihood Function¶

Location: Create a new function following the naming pattern {model_name}_logp_inner_func

Purpose: Define the core computational logic for your RLSSM model. See the existing implementation in rldm.py for template details.

Details:

• Function signature: Must follow the pattern:

  def my_custom_rlssm_logp_inner_func(
      subj,
      ntrials_subj,
      data,
      *model_params,    # Your specific parameters
      *extra_fields,    # Additional data fields
  ):
  

• Input requirements:

  • subj: Subject index
  • ntrials_subj: Number of trials per subject
  • data: RT and response data matrix
  • Parameters must match the order specified in list_params
  • Extra fields must match those specified in extra_fields

• Output requirements:

  • Return a 1D array of log likelihoods for each trial
  • Must be differentiable for gradient-based sampling/optimization

• Implementation notes:

  • Use JAX operations for automatic differentiation
  • Handle parameter slicing for individual subjects using dynamic_slice
  • Implement your specific RL update rules and decision mechanisms
  • Structure the LAN matrix according to your decision model requirements

Step 4: Update the Vectorized Function Reference¶

Location: Modify the rldm_logp_inner_func_vmapped assignment

Purpose: Enable parallel computation across multiple subjects.

Details:

• Update the function reference to point to your custom likelihood function
• The vectorization pattern remains the same - only the first argument (subject index) gets vectorized
• Ensure the total_params count includes all parameters plus extra fields plus data columns

Example:

rldm_logp_inner_func_vmapped = jax.vmap(
    my_custom_rlssm_logp_inner_func,  # Update this to your function name
    in_axes=[0] + [None] * total_params,
)

Step 5: Update Parameter Unpacking in the Main Likelihood Function¶

Location: Modify the logp function within make_logp_func

Purpose: Ensure parameters are correctly extracted and passed to your custom function.

Details:

• Parameter extraction: Adjust the indexing to match your model's parameter structure:

  # Extract extra fields (adjust indices based on your model)
  participant_id = dist_params[n_params]      # Usually after all model params
  trial = dist_params[n_params + 1]
  feedback = dist_params[n_params + 2]
  # ... additional extra fields as needed
  
  # Extract model parameters
  param1, param2, ..., paramN = dist_params[:MODEL_CONFIG["n_params"]]
  

• Function call: Update the vec_logp call to pass parameters in the correct order:

  return vec_logp(
      subj,
      n_trials,
      data,
      param1,    # Your specific parameters
      param2,
      # ...
      paramN,
      trial,     # Extra fields
      feedback,
      # ...
  )
  

Step 6: Verify VJP Function Compatibility¶

Location: Check the vjp_logp function within make_vjp_logp_func

Purpose: Ensure gradient computation works correctly with your parameter structure.

Details:

• The VJP (Vector-Jacobian Product) function should automatically work with your custom model
• The slicing [1:MODEL_CONFIG["n_params"] + 1] excludes the data and extra fields from gradient computation
• No changes typically needed unless you have special gradient requirements

Notes:

• VJP is used for automatic differentiation during MCMC sampling
• The function computes gradients with respect to model parameters only
• Extra fields (trial, feedback, etc.) are not differentiated
• If your model has unusual parameter dependencies, you may need custom gradient handling

Section II: Using the custom model with HSSM¶

Load and prepare the demo dataset¶

This data file contains (synthetic) data from a simulated 2-armed bandit task. We examine the dataset -- it contains the typical columns that are expected from a canonical instrumental learning task. participant_id identifies the subject id, trial identifies the sequence of trials within the subject data, response and rt are the data columns recorded for each trial, feedback column shows the reward obtained on a given trial and correct records whether the response was correct.

In [3]:

# Load synthetic RLSSM dataset containing both behavioral data and ground truth parameters
savefile = np.load("../../tests/fixtures/rldm_data.npy", allow_pickle=True).item()
dataset = savefile['data']

# Rename trial column to match HSSM conventions
dataset.rename(columns={'trial': 'trial_id'}, inplace=True)

# Examine the dataset structure
dataset.head()

# Load synthetic RLSSM dataset containing both behavioral data and ground truth parameters savefile = np.load("../../tests/fixtures/rldm_data.npy", allow_pickle=True).item() dataset = savefile['data'] # Rename trial column to match HSSM conventions dataset.rename(columns={'trial': 'trial_id'}, inplace=True) # Examine the dataset structure dataset.head()

Out[3]:

	participant_id	trial_id	response	rt	feedback	correct
0	0	0	0.0	0.935602	0.126686	0.0
1	0	1	0.0	1.114379	0.173100	0.0
2	0	2	0.0	0.564311	0.444935	0.0
3	0	3	0.0	2.885860	0.307207	0.0
4	0	4	0.0	0.532113	0.177911	0.0

In [4]:

# Validate data structure and extract dataset configuration 
dataset, n_participants, n_trials = hssm.check_data_for_rl(dataset)

print(f"Number of participants: {n_participants}")
print(f"Number of trials: {n_trials}")

# Validate data structure and extract dataset configuration dataset, n_participants, n_trials = hssm.check_data_for_rl(dataset) print(f"Number of participants: {n_participants}") print(f"Number of trials: {n_trials}")

Number of participants: 20
Number of trials: 200

Construct HSSM-compatible PyMC distribution from a simulator and JAX likelihood callable¶

We now construct a custom model that is compatible with HSSM and PyMC. Note that HSSM internally constructs a PyMC object (which is used for sampling) based on the user-specified HSSM model. In other words, we are peeling the abstration layers conveniently afforded by HSSM to directly use the core machinery of HSSM. This advanced HSSM tutorial explains how to use HSSM when starting from the very basics of a model -- a simulator and a JAX likelihood callable.

The simulator function is used for generating samples from the model (for posterior predictives, etc.) and the likelihood callable is employed for sampling/inference. This preview tutorial exposes the key flexibility of the HSSM for use in fitting RLSSM models. Therefore, the subsequent tutorial will focus only on the sampling/inference aspect. We create a dummy simulator function to bypass the need for defining the actual simulator.

Step 1: Define a pytensor RandomVariable¶

In [5]:

# Define parameters for the RLSSM model (RL + decision model parameters)
list_params = ['rl.alpha', 'rl.alpha_neg', 'scaler', 'a', 'z', 't', 'theta']

# Create a dummy simulator for generating synthetic data (used for posterior predictives)
# This bypasses the need for a full RLSSM simulator implementation
def create_dummy_simulator():
    """Create a dummy simulator function for RLSSM model."""
    def sim_wrapper(simulator_fun, theta, model, n_samples, random_state, **kwargs):
        # Generate random RT and choice data as placeholders
        sim_rt = np.random.uniform(0.2, 0.6, n_samples)
        sim_ch = np.random.randint(0, 2, n_samples)
        
        return np.column_stack([sim_rt, sim_ch])

    # Wrap the simulator function with required metadata
    wrapped_simulator = partial(sim_wrapper, simulator_fun=simulator, model="custom", n_samples=1)

    # Decorate the simulator to make it compatible with HSSM
    return decorate_atomic_simulator(model_name="custom", choices=[0, 1], obs_dim=2)(wrapped_simulator)

# Create the simulator and RandomVariable
decorated_simulator = create_dummy_simulator()

# Create a PyTensor RandomVariable using `make_hssm_rv` for use in the PyMC model
CustomRV = make_hssm_rv(
    simulator_fun=decorated_simulator, list_params=list_params
)

# Define parameters for the RLSSM model (RL + decision model parameters) list_params = ['rl.alpha', 'rl.alpha_neg', 'scaler', 'a', 'z', 't', 'theta'] # Create a dummy simulator for generating synthetic data (used for posterior predictives) # This bypasses the need for a full RLSSM simulator implementation def create_dummy_simulator(): """Create a dummy simulator function for RLSSM model.""" def sim_wrapper(simulator_fun, theta, model, n_samples, random_state, **kwargs): # Generate random RT and choice data as placeholders sim_rt = np.random.uniform(0.2, 0.6, n_samples) sim_ch = np.random.randint(0, 2, n_samples) return np.column_stack([sim_rt, sim_ch]) # Wrap the simulator function with required metadata wrapped_simulator = partial(sim_wrapper, simulator_fun=simulator, model="custom", n_samples=1) # Decorate the simulator to make it compatible with HSSM return decorate_atomic_simulator(model_name="custom", choices=[0, 1], obs_dim=2)(wrapped_simulator) # Create the simulator and RandomVariable decorated_simulator = create_dummy_simulator() # Create a PyTensor RandomVariable using `make_hssm_rv` for use in the PyMC model CustomRV = make_hssm_rv( simulator_fun=decorated_simulator, list_params=list_params )

Step 2: Define a likelihood function¶

In [6]:

# Create a Pytensor Op for the likelihood function.
# The `make_rldm_logp_op` function is a utility that wraps the base JAX likelihood function into a HSSM/PyMC-compatible callable.

logp_jax_op = make_rldm_logp_op(
    n_participants=n_participants,
    n_trials=n_trials,
    n_params=len(list_params),
)

# Create a Pytensor Op for the likelihood function. # The `make_rldm_logp_op` function is a utility that wraps the base JAX likelihood function into a HSSM/PyMC-compatible callable. logp_jax_op = make_rldm_logp_op( n_participants=n_participants, n_trials=n_trials, n_params=len(list_params), )

In [7]:

# Test the likelihood function

def extract_data_columns(dataset):
    """Extract required data columns from dataset."""
    return {
        'participant_id': dataset["participant_id"].values,
        'trial': dataset["trial_id"].values,
        'response': dataset["response"].values,
        'feedback': dataset["feedback"].values,
        'rt': dataset["rt"].values
    }

def create_test_parameters(n_trials):
    """Create dummy parameters for testing the likelihood function."""
    return {
        'rl_alpha': np.ones(n_trials) * 0.60,
        'rl_alpha_neg': np.ones(n_trials) * 0.60,  
        'scaler': np.ones(n_trials) * 3.2,
        'a': np.ones(n_trials) * 1.2,
        'z': np.ones(n_trials) * 0.1,
        't': np.ones(n_trials) * 0.1,
        'theta': np.ones(n_trials) * 0.1
}

# Extract data and create test parameters
data_columns = extract_data_columns(dataset)
num_subj = len(np.unique(data_columns['participant_id']))
n_trials_total = num_subj * 200

test_params = create_test_parameters(n_trials_total)

# Evaluate the likelihood function
test_logp_out = logp_jax_op(
    np.column_stack((data_columns['rt'], data_columns['response'])),
    test_params['rl_alpha'],
    test_params['rl_alpha_neg'],
    test_params['scaler'],
    test_params['a'], 
    test_params['z'],
    test_params['t'],
    test_params['theta'],
    data_columns['participant_id'],
    data_columns['trial'],
    data_columns['feedback'],
)

LL = test_logp_out.eval()
print(f"Log likelihood: {np.sum(LL):.4f}")

# Test the likelihood function def extract_data_columns(dataset): """Extract required data columns from dataset.""" return { 'participant_id': dataset["participant_id"].values, 'trial': dataset["trial_id"].values, 'response': dataset["response"].values, 'feedback': dataset["feedback"].values, 'rt': dataset["rt"].values } def create_test_parameters(n_trials): """Create dummy parameters for testing the likelihood function.""" return { 'rl_alpha': np.ones(n_trials) * 0.60, 'rl_alpha_neg': np.ones(n_trials) * 0.60, 'scaler': np.ones(n_trials) * 3.2, 'a': np.ones(n_trials) * 1.2, 'z': np.ones(n_trials) * 0.1, 't': np.ones(n_trials) * 0.1, 'theta': np.ones(n_trials) * 0.1 } # Extract data and create test parameters data_columns = extract_data_columns(dataset) num_subj = len(np.unique(data_columns['participant_id'])) n_trials_total = num_subj * 200 test_params = create_test_parameters(n_trials_total) # Evaluate the likelihood function test_logp_out = logp_jax_op( np.column_stack((data_columns['rt'], data_columns['response'])), test_params['rl_alpha'], test_params['rl_alpha_neg'], test_params['scaler'], test_params['a'], test_params['z'], test_params['t'], test_params['theta'], data_columns['participant_id'], data_columns['trial'], data_columns['feedback'], ) LL = test_logp_out.eval() print(f"Log likelihood: {np.sum(LL):.4f}")

Log likelihood: -6879.1526

Step 3: Define a model config and HSSM model¶

In [8]:

# Step 3: Define the model config

# Configure the HSSM model 
model_config = hssm.ModelConfig(
    response=["rt", "response"],        # Dependent variables (RT and choice)
    list_params=                        # List of model parameters
        ['rl.alpha', 'rl.alpha_neg', 'scaler', 'a', 'z', 't', 'theta'],            
    choices=[0, 1],                     # Possible choice options
    default_priors={},                  # Use custom priors (defined below)
    bounds=dict(                        # Parameter bounds for optimization
        rl_alpha=(0.01, 1),             # Learning rate bounds
        rl_alpha_neg=(0.01, 1),         # Negative learning rate bounds
        scaler=(1, 4),                  # Scaler bounds
        a=(0.3, 2.5),                   # Boundary separation bounds
        z=(0.1, 0.9),                   # Bias bounds
        t=(0.1, 2.0),                   # Non-decision time bounds
        theta=(0.0, 1.2)                # Collapse rate bounds
        ),
    rv=CustomRV,                        # Custom RandomVariable that we created earlier
    extra_fields=[                      # Additional data columns to be passed to the likelihood function as extra_fields
        "participant_id", 
        "trial_id", 
        "feedback"],  
    backend="jax"                       # Use JAX for computation
)

# Step 3: Define the model config # Configure the HSSM model model_config = hssm.ModelConfig( response=["rt", "response"], # Dependent variables (RT and choice) list_params= # List of model parameters ['rl.alpha', 'rl.alpha_neg', 'scaler', 'a', 'z', 't', 'theta'], choices=[0, 1], # Possible choice options default_priors={}, # Use custom priors (defined below) bounds=dict( # Parameter bounds for optimization rl_alpha=(0.01, 1), # Learning rate bounds rl_alpha_neg=(0.01, 1), # Negative learning rate bounds scaler=(1, 4), # Scaler bounds a=(0.3, 2.5), # Boundary separation bounds z=(0.1, 0.9), # Bias bounds t=(0.1, 2.0), # Non-decision time bounds theta=(0.0, 1.2) # Collapse rate bounds ), rv=CustomRV, # Custom RandomVariable that we created earlier extra_fields=[ # Additional data columns to be passed to the likelihood function as extra_fields "participant_id", "trial_id", "feedback"], backend="jax" # Use JAX for computation )

In [9]:

# Create a hierarchical HSSM model with custom likelihood function
hssm_model = hssm.HSSM(
    data=dataset,                        # Input dataset
    model_config=model_config,           # Model configuration
    p_outlier=0,                         # No outlier modeling
    lapse=None,                          # No lapse rate modeling
    loglik=logp_jax_op,                  # Custom RLDM likelihood function
    loglik_kind="approx_differentiable", # Use approximate gradients
    noncentered=True,                    # Use non-centered parameterization
    process_initvals=False,              # Skip initial value processing in HSSM
    include=[
        # Define hierarchical priors: group-level intercepts + subject-level random effects
        hssm.Param("rl.alpha", 
                formula="rl_alpha ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=1, mu=0.3)}),
        hssm.Param("rl.alpha_neg", 
                formula="rl_alpha_neg ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=1, mu=0.3)}),
        hssm.Param("scaler", 
                formula="scaler ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=1, upper=4, mu=1.5)}),
        hssm.Param("a", 
                formula="a ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.3, upper=2.5, mu=1.0)}),
        hssm.Param("z", 
                formula="z ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.1, upper=0.9, mu=0.2)}),
        hssm.Param("t", 
                formula="t ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=2, mu=0.2, initval=0.1)}),
        hssm.Param("theta", 
                formula="theta ~ 1 + (1|participant_id)", 
                prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.00, upper=1.2, mu=0.3)}),
    ]
)

# Create a hierarchical HSSM model with custom likelihood function hssm_model = hssm.HSSM( data=dataset, # Input dataset model_config=model_config, # Model configuration p_outlier=0, # No outlier modeling lapse=None, # No lapse rate modeling loglik=logp_jax_op, # Custom RLDM likelihood function loglik_kind="approx_differentiable", # Use approximate gradients noncentered=True, # Use non-centered parameterization process_initvals=False, # Skip initial value processing in HSSM include=[ # Define hierarchical priors: group-level intercepts + subject-level random effects hssm.Param("rl.alpha", formula="rl_alpha ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=1, mu=0.3)}), hssm.Param("rl.alpha_neg", formula="rl_alpha_neg ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=1, mu=0.3)}), hssm.Param("scaler", formula="scaler ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=1, upper=4, mu=1.5)}), hssm.Param("a", formula="a ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.3, upper=2.5, mu=1.0)}), hssm.Param("z", formula="z ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.1, upper=0.9, mu=0.2)}), hssm.Param("t", formula="t ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.01, upper=2, mu=0.2, initval=0.1)}), hssm.Param("theta", formula="theta ~ 1 + (1|participant_id)", prior={"Intercept": hssm.Prior("TruncatedNormal", lower=0.00, upper=1.2, mu=0.3)}), ] )

No common intercept. Bounds for parameter rl.alpha is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter rl.alpha_neg is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter scaler is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter a is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter z is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter t is not applied due to a current limitation of Bambi. This will change in the future.
No common intercept. Bounds for parameter theta is not applied due to a current limitation of Bambi. This will change in the future.
Model initialized successfully.

Sample using NUTS MCMC¶

In [10]:

# Run MCMC sampling using NUTS sampler with JAX backend
# Note: Using small number of samples for demonstration (increase for real analysis)
idata_mcmc = hssm_model.sample(
    sampler='nuts_numpyro',  # JAX-based NUTS sampler for efficiency
    chains=1,                # Number of parallel chains
    draws=1000,                # Number of posterior samples
    tune=1000,                 # Number of tuning/warmup samples
)

# Run MCMC sampling using NUTS sampler with JAX backend # Note: Using small number of samples for demonstration (increase for real analysis) idata_mcmc = hssm_model.sample( sampler='nuts_numpyro', # JAX-based NUTS sampler for efficiency chains=1, # Number of parallel chains draws=1000, # Number of posterior samples tune=1000, # Number of tuning/warmup samples )

Using default initvals. 

sample: 100%|██████████| 2000/2000 [08:33<00:00,  3.89it/s, 29 steps of size 1.56e-01. acc. prob=0.92]   
There were 1 divergences after tuning. Increase `target_accept` or reparameterize.
Only one chain was sampled, this makes it impossible to run some convergence checks
/Users/krishnbera/Documents/revert_rldm/HSSM/.venv/lib/python3.12/site-packages/pymc/pytensorf.py:958: FutureWarning: compile_pymc was renamed to compile. Old name will be removed in a future release of PyMC
  warnings.warn(
100%|██████████| 1000/1000 [00:02<00:00, 366.61it/s]

In [11]:

idata_mcmc

idata_mcmc

Out[11]:

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 2MB
  Dimensions:                                (chain: 1, draw: 1000,
                                              participant_id__factor_dim: 20,
                                              rl.alpha_1|participant_id__factor_dim: 20)
  Coordinates:
    * chain                                  (chain) int64 8B 0
    * draw                                   (draw) int64 8kB 0 1 2 ... 998 999
    * rl.alpha_1|participant_id__factor_dim  (rl.alpha_1|participant_id__factor_dim) <U2 160B ...
    * participant_id__factor_dim             (participant_id__factor_dim) <U2 160B ...
  Data variables: (12/35)
      rl.alpha_neg_1|participant_id          (chain, draw, participant_id__factor_dim) float64 160kB ...
      rl.alpha_neg_1|participant_id_sigma    (chain, draw) float64 8kB 0.09328 ...
      a_1|participant_id_mu                  (chain, draw) float64 8kB -0.2027 ...
      rl.alpha_1|participant_id_sigma        (chain, draw) float64 8kB 0.03706 ...
      scaler_1|participant_id_offset         (chain, draw, participant_id__factor_dim) float64 160kB ...
      a_1|participant_id                     (chain, draw, participant_id__factor_dim) float64 160kB ...
      ...                                     ...
      theta_1|participant_id_mu              (chain, draw) float64 8kB -0.2717 ...
      a_Intercept                            (chain, draw) float64 8kB 1.491 .....
      scaler_Intercept                       (chain, draw) float64 8kB 2.639 .....
      scaler_1|participant_id_mu             (chain, draw) float64 8kB 0.3354 ....
      rl.alpha_1|participant_id_mu           (chain, draw) float64 8kB 0.02453 ...
      theta_1|participant_id                 (chain, draw, participant_id__factor_dim) float64 160kB ...
  Attributes:
      created_at:                  2025-07-15T19:55:45.042734+00:00
      arviz_version:               0.22.0
      inference_library:           numpyro
      inference_library_version:   0.18.0
      sampling_time:               517.168047
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 1000
    • participant_id__factor_dim: 20
    • rl.alpha_1|participant_id__factor_dim: 20
  • Coordinates: (4)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999], shape=(1000,))

    • rl.alpha_1|participant_id__factor_dim
      (rl.alpha_1|participant_id__factor_dim)
      <U2
      '0' '1' '2' '3' ... '17' '18' '19'

      array(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12',
             '13', '14', '15', '16', '17', '18', '19'], dtype='<U2')

    • participant_id__factor_dim
      (participant_id__factor_dim)
      <U2
      '0' '1' '2' '3' ... '17' '18' '19'

      array(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12',
             '13', '14', '15', '16', '17', '18', '19'], dtype='<U2')

  • Data variables: (35)
    • rl.alpha_neg_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      0.08525 -0.03887 ... 0.1017 0.1859

      array([[[ 8.52524246e-02, -3.88736108e-02,  1.10405443e-01, ...,
               -6.03191902e-02,  1.11141755e-02,  9.78937925e-02],
              [ 1.25520353e-02, -6.54906468e-03, -1.25774425e-02, ...,
               -5.00059581e-04,  4.13463470e-03,  9.16094207e-04],
              [ 7.28220291e-05, -3.21929838e-03, -6.87044825e-03, ...,
                2.58669913e-04,  5.35369655e-03,  5.29310912e-03],
              ...,
              [ 6.19613710e-02, -9.04301011e-02, -6.40939936e-02, ...,
               -1.44371076e-01,  6.93436508e-02,  1.19720281e-01],
              [ 1.17282010e-01, -4.68099183e-02, -5.80185783e-02, ...,
                8.10048782e-02,  6.91800987e-02,  5.62488894e-02],
              [ 4.16863398e-02,  4.87059201e-02, -8.28223216e-02, ...,
                1.12394939e-01,  1.01691578e-01,  1.85925093e-01]]],
            shape=(1, 1000, 20))

    • rl.alpha_neg_1|participant_id_sigma
      (chain, draw)
      float64
      0.09328 0.01113 ... 0.0792 0.08951

      array([[0.09327861, 0.01113374, 0.0067807 , 0.02224991, 0.18818852,
              0.13693416, 0.0137104 , 0.12112734, 0.02774661, 0.01184964,
              0.03313942, 0.10921727, 0.09487614, 0.01795892, 0.10265819,
              0.06051296, 0.08517624, 0.08433337, 0.11109638, 0.06801696,
              0.02649293, 0.07327044, 0.10900929, 0.1437742 , 0.11890193,
              0.01861684, 0.03742197, 0.16033764, 0.0679923 , 0.11418525,
              0.12090175, 0.06131621, 0.07123911, 0.09237216, 0.03997358,
              0.02860808, 0.08295873, 0.13752383, 0.07696958, 0.0946859 ,
              0.06441234, 0.0692772 , 0.03525691, 0.14056736, 0.03736379,
              0.14179708, 0.00567365, 0.07516111, 0.0841962 , 0.18274141,
              0.0527435 , 0.04581121, 0.01990324, 0.05346567, 0.16033973,
              0.06810372, 0.10893175, 0.16168255, 0.03397938, 0.02351622,
              0.08627316, 0.01601121, 0.01131003, 0.01035117, 0.02602841,
              0.07322761, 0.03045117, 0.05172015, 0.05411817, 0.00919667,
              0.08180663, 0.01927885, 0.06548393, 0.10205384, 0.0270732 ,
              0.00521116, 0.02167259, 0.08553463, 0.01201813, 0.21275623,
              0.06559833, 0.1847809 , 0.05703139, 0.0461503 , 0.03027838,
              0.21694742, 0.12979277, 0.11097162, 0.08269843, 0.04748068,
              0.28007182, 0.0231027 , 0.02844314, 0.15013276, 0.17306386,
              0.10448734, 0.02659661, 0.14779651, 0.03827021, 0.11011419,
      ...
              0.03162604, 0.08527373, 0.12920783, 0.06979845, 0.10271215,
              0.06767155, 0.03451224, 0.15729287, 0.11459939, 0.04799843,
              0.03906045, 0.10477396, 0.06634792, 0.04507225, 0.211674  ,
              0.01846484, 0.10071775, 0.02165307, 0.10840669, 0.04564035,
              0.07983719, 0.1140913 , 0.14494868, 0.04455192, 0.02871864,
              0.06410322, 0.07585762, 0.06543017, 0.09677864, 0.08256108,
              0.07546982, 0.08211627, 0.03127799, 0.1853499 , 0.07410288,
              0.03271516, 0.03283377, 0.09298921, 0.08625752, 0.04118969,
              0.11363661, 0.11267627, 0.11052486, 0.10924007, 0.07988203,
              0.07742597, 0.11399736, 0.08374279, 0.08072191, 0.1551673 ,
              0.17327917, 0.07352062, 0.12750346, 0.09591898, 0.03928874,
              0.08258566, 0.19549517, 0.05626114, 0.12524791, 0.08005122,
              0.11528304, 0.08485011, 0.07287171, 0.00884009, 0.10423099,
              0.06677659, 0.0590118 , 0.14027077, 0.11099063, 0.11157492,
              0.16431369, 0.16567496, 0.09144268, 0.01535024, 0.01808003,
              0.01111863, 0.17841836, 0.01905178, 0.1746642 , 0.10077163,
              0.1998371 , 0.11643685, 0.11606395, 0.06974784, 0.06793704,
              0.02075596, 0.05559657, 0.12647182, 0.02907609, 0.10676777,
              0.0145071 , 0.0222392 , 0.0448217 , 0.10051719, 0.0922675 ,
              0.06733797, 0.12014162, 0.13143749, 0.07920041, 0.08951135]])

    • a_1|participant_id_mu
      (chain, draw)
      float64
      -0.2027 -0.4448 ... -0.1654 -0.1497

      array([[-2.02719840e-01, -4.44842080e-01, -3.03964924e-01,
              -1.55609758e-01,  1.31007984e-01, -7.59249147e-02,
               1.37072883e-01,  5.06277206e-02, -5.60636524e-02,
               1.84903475e-01,  2.41388470e-01,  8.32442934e-02,
              -4.56799516e-02,  1.33342697e-02, -1.42448090e-01,
               6.53218550e-02,  1.71898816e-01, -1.09097340e-02,
               3.98032202e-01, -2.99370619e-01,  3.05871182e-01,
              -2.50432920e-01,  3.32773988e-01, -2.64016270e-01,
               2.68458099e-01, -4.70057178e-01, -4.73121869e-01,
               1.10616487e-01, -1.11512175e-01,  7.82328434e-02,
               1.68435968e-01, -3.64161262e-01, -1.79503239e-02,
               4.90399166e-01, -3.68400984e-01, -8.57769115e-02,
               1.15795612e-02,  6.29667855e-03, -2.04432032e-01,
               2.02540726e-01, -3.54064121e-01, -1.86987814e-01,
              -1.78315042e-01,  4.05421330e-01, -6.33678532e-01,
              -7.59063197e-03, -2.27643846e-01,  1.76635714e-01,
              -4.64064271e-02,  1.08984697e-01, -1.31863439e-01,
              -2.65443662e-01, -3.32474608e-01,  2.07935384e-01,
              -3.40245641e-01,  4.48081785e-02,  2.40600620e-01,
              -7.65335512e-02,  8.40249994e-02,  4.46943186e-02,
      ...
              -5.25219200e-02,  3.26874443e-02, -5.35956329e-03,
               2.64440432e-03,  2.68161531e-01, -3.80900520e-01,
              -3.63901515e-01,  4.43749513e-01, -2.62191913e-01,
               4.85944373e-01, -3.73407644e-01,  3.08463063e-01,
               3.08789186e-01, -3.26877636e-01,  7.66973819e-02,
              -2.03142913e-01,  4.00175991e-01, -3.99422717e-01,
               2.18065014e-01, -2.91034906e-01,  1.31350513e-01,
               1.45869259e-01, -1.04896259e-01,  7.87857837e-02,
               1.28148804e-01,  2.08061623e-01, -2.28293052e-01,
               1.30602673e-01, -3.26786851e-01,  2.81410262e-01,
              -3.34005531e-01,  2.84284392e-01, -3.80217659e-01,
              -3.00765996e-01,  2.81146102e-01, -2.59827388e-01,
               1.67198897e-01, -1.89070258e-01, -2.87456083e-01,
               1.16248583e-01, -1.13738792e-01,  1.33560394e-01,
              -1.66193673e-01, -2.09997026e-01,  2.52654322e-01,
               5.11441764e-02, -1.58902464e-01,  1.61687130e-01,
              -1.83569790e-01,  1.10535873e-01, -2.45399914e-01,
               3.78714947e-01, -3.73786765e-01,  4.98948500e-01,
               2.03085198e-01, -4.68995390e-02, -1.65352499e-01,
              -1.49680850e-01]])

    • rl.alpha_1|participant_id_sigma
      (chain, draw)
      float64
      0.03706 0.03513 ... 0.06522 0.02886

      array([[0.03705929, 0.03513307, 0.02895748, 0.06591059, 0.16732787,
              0.04916698, 0.15706138, 0.06801485, 0.09469248, 0.14255108,
              0.12811558, 0.22328067, 0.05717146, 0.14317605, 0.15689983,
              0.01769085, 0.02674053, 0.02704576, 0.02184269, 0.01961838,
              0.1458844 , 0.05309886, 0.10834505, 0.1546012 , 0.0312736 ,
              0.07673636, 0.07006105, 0.13105068, 0.11640804, 0.05656212,
              0.06250989, 0.09320703, 0.12368692, 0.01581294, 0.07802086,
              0.17113857, 0.09322637, 0.03732654, 0.12476267, 0.12650323,
              0.0889501 , 0.11217427, 0.11363395, 0.22390902, 0.03064159,
              0.02317336, 0.16881594, 0.03566653, 0.11702845, 0.12819443,
              0.06096566, 0.08023278, 0.06755874, 0.06227903, 0.10302292,
              0.06773351, 0.07629667, 0.1396328 , 0.19289015, 0.13320769,
              0.04224377, 0.05855081, 0.22075669, 0.0448352 , 0.13730017,
              0.03361158, 0.04601898, 0.23437392, 0.08898296, 0.029606  ,
              0.07635805, 0.14949171, 0.08975536, 0.04087383, 0.08667208,
              0.10378602, 0.05187667, 0.14663556, 0.01525829, 0.09895022,
              0.1219454 , 0.03420199, 0.16709584, 0.07488889, 0.11075151,
              0.11335392, 0.09615191, 0.13245751, 0.04418025, 0.10621877,
              0.02593165, 0.07919483, 0.1304459 , 0.01230364, 0.00486108,
              0.21220025, 0.12718461, 0.12606474, 0.11468334, 0.10354153,
      ...
              0.14117324, 0.07014331, 0.12533277, 0.06199117, 0.14506578,
              0.15604541, 0.03515604, 0.20747414, 0.03717839, 0.02070695,
              0.11343202, 0.03029284, 0.04269185, 0.03996101, 0.01773622,
              0.07836577, 0.05189516, 0.21696966, 0.05786292, 0.05962564,
              0.08947374, 0.04780684, 0.13209795, 0.06296797, 0.18522729,
              0.12441249, 0.03614721, 0.21138901, 0.10233696, 0.10215774,
              0.04820891, 0.16667374, 0.17328293, 0.03794996, 0.07960459,
              0.11472293, 0.08932149, 0.01794318, 0.04524802, 0.07987832,
              0.18342589, 0.0048896 , 0.00530082, 0.26090563, 0.02460495,
              0.0313102 , 0.17797002, 0.01851477, 0.18591341, 0.08829196,
              0.04356739, 0.07960325, 0.12178655, 0.01614376, 0.10168777,
              0.02288544, 0.13287343, 0.05188245, 0.12853207, 0.12999807,
              0.13337968, 0.03015042, 0.09442344, 0.08705122, 0.08415245,
              0.17540988, 0.15582279, 0.14589611, 0.02524112, 0.09966723,
              0.08574016, 0.16832156, 0.119172  , 0.052289  , 0.19453834,
              0.05346085, 0.08721493, 0.09342031, 0.07427953, 0.09782203,
              0.08154349, 0.02543633, 0.27488179, 0.04897705, 0.09932784,
              0.2270581 , 0.10833157, 0.04251109, 0.11566376, 0.07376173,
              0.0549153 , 0.16695815, 0.04692031, 0.04356361, 0.13768075,
              0.04178491, 0.07694743, 0.08435779, 0.06521522, 0.02886193]])

    • scaler_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      0.05816 0.8928 ... -0.5987 -0.07629

      array([[[ 0.05815785,  0.89281459, -1.72070993, ...,  0.0870186 ,
               -1.0537266 , -2.68888698],
              [-0.54158608,  0.96019242, -0.85502858, ...,  0.32913185,
               -0.48634642, -1.51089929],
              [ 0.2002351 ,  1.11554375, -0.15931935, ...,  0.66193168,
               -1.16415439, -0.85202308],
              ...,
              [-1.12131754,  1.50988892, -0.0566591 , ...,  0.00762951,
               -0.30476515, -1.2250236 ],
              [-0.35235071,  0.14721624,  0.04704862, ..., -0.48884411,
               -0.74038562, -0.26226789],
              [-0.64990326,  1.41503762,  0.45225491, ..., -0.59423638,
               -0.59874432, -0.07628799]]], shape=(1, 1000, 20))

    • a_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      -0.01662 -0.03004 ... 0.01428

      array([[[-0.01662183, -0.03004255, -0.01417413, ...,  0.12101847,
               -0.00584561,  0.0608771 ],
              [ 0.08242411,  0.02024621,  0.03138023, ...,  0.11866479,
               -0.01507013,  0.00681003],
              [ 0.04010624,  0.05536735,  0.04770319, ...,  0.08566125,
               -0.04251608,  0.00275873],
              ...,
              [ 0.02546625,  0.02139674,  0.02826233, ..., -0.02260432,
                0.01751147,  0.05382402],
              [ 0.00293344, -0.02290545,  0.02147935, ...,  0.02181695,
                0.00510547,  0.0131984 ],
              [ 0.0007985 , -0.00398267,  0.02638743, ...,  0.03311368,
               -0.00804651,  0.01428182]]], shape=(1, 1000, 20))

    • z_1|participant_id_sigma
      (chain, draw)
      float64
      0.00492 0.005791 ... 0.02391

      array([[0.00491989, 0.00579056, 0.0074441 , 0.00643979, 0.00245682,
              0.00543548, 0.01248916, 0.00607091, 0.00719514, 0.00150427,
              0.00170021, 0.00801611, 0.00427387, 0.00092616, 0.02129892,
              0.02307831, 0.02231576, 0.01088997, 0.01520426, 0.01160025,
              0.00274968, 0.0089091 , 0.01775793, 0.01624618, 0.01101128,
              0.00575512, 0.00861218, 0.02051636, 0.01270545, 0.01989488,
              0.03869494, 0.02960841, 0.00713568, 0.02938418, 0.0086812 ,
              0.00578054, 0.01448172, 0.00844005, 0.02639904, 0.01021372,
              0.0107375 , 0.03207331, 0.03313816, 0.01127482, 0.00638755,
              0.0071244 , 0.0058769 , 0.00612335, 0.00382838, 0.00726529,
              0.02161403, 0.01047378, 0.02329702, 0.01367762, 0.01836122,
              0.01663674, 0.01937311, 0.01342901, 0.01455493, 0.01139901,
              0.02275929, 0.01347695, 0.00573815, 0.00083742, 0.00023218,
              0.00708994, 0.00944464, 0.00590824, 0.02266395, 0.01884882,
              0.01852673, 0.01424532, 0.02465111, 0.01469275, 0.01661927,
              0.00711061, 0.01658726, 0.02262509, 0.01039266, 0.00839315,
              0.00284973, 0.00836196, 0.03001547, 0.02045738, 0.00317558,
              0.0064545 , 0.00987383, 0.01704496, 0.02013295, 0.00463856,
              0.02519563, 0.00755671, 0.01060129, 0.01790545, 0.0139149 ,
              0.01389597, 0.00696438, 0.00936576, 0.02195523, 0.0264061 ,
      ...
              0.00230654, 0.00212305, 0.01871171, 0.01251532, 0.0124612 ,
              0.01696136, 0.02007865, 0.01339983, 0.01314714, 0.01010116,
              0.01825203, 0.02100792, 0.01221691, 0.01717949, 0.00870647,
              0.02607905, 0.01363287, 0.03002605, 0.01936816, 0.0145294 ,
              0.01312378, 0.00750425, 0.00696661, 0.00690587, 0.01508316,
              0.0071477 , 0.02344095, 0.01139574, 0.00398835, 0.0156852 ,
              0.00407706, 0.00426644, 0.00634134, 0.00804195, 0.01828475,
              0.01550731, 0.01108918, 0.01346289, 0.02826007, 0.03568245,
              0.02140006, 0.02756971, 0.02731618, 0.02175558, 0.02316891,
              0.02143751, 0.01973289, 0.0271815 , 0.03375149, 0.02418228,
              0.04339554, 0.02298121, 0.03133625, 0.01931691, 0.02477765,
              0.00344268, 0.01426477, 0.00303824, 0.00545446, 0.00512166,
              0.00321535, 0.01226526, 0.02597917, 0.00764796, 0.01065695,
              0.02250729, 0.02525534, 0.02066713, 0.01133463, 0.01191164,
              0.01174052, 0.00282082, 0.00239297, 0.00751036, 0.01456332,
              0.00919505, 0.00679071, 0.01195801, 0.01285645, 0.00935799,
              0.01963981, 0.00911502, 0.02673968, 0.0133325 , 0.02642059,
              0.0164162 , 0.02801497, 0.01176376, 0.00947875, 0.00273043,
              0.00635398, 0.01679194, 0.01717792, 0.00051236, 0.01394578,
              0.02879991, 0.02321684, 0.01631488, 0.00777396, 0.02390892]])

    • theta_Intercept
      (chain, draw)
      float64
      0.3538 0.3587 ... 0.3633 0.3621

      array([[0.35378513, 0.35871432, 0.36349141, 0.3569534 , 0.38645104,
              0.37364719, 0.39190438, 0.34870226, 0.39528835, 0.36503875,
              0.37380307, 0.39015077, 0.39216443, 0.38502772, 0.36366708,
              0.38967961, 0.40475299, 0.39391405, 0.39148225, 0.37241536,
              0.36125535, 0.37056421, 0.37421543, 0.38330909, 0.37828627,
              0.36118888, 0.37825735, 0.37120072, 0.38070178, 0.37825277,
              0.37077443, 0.38020013, 0.3791562 , 0.35011478, 0.37423386,
              0.37471018, 0.3754635 , 0.3807245 , 0.38123446, 0.34858397,
              0.39979103, 0.38171391, 0.3901461 , 0.37183075, 0.37548151,
              0.36752798, 0.39265423, 0.37485955, 0.37652029, 0.37007421,
              0.3694082 , 0.38591065, 0.38717194, 0.41618991, 0.37016173,
              0.37663102, 0.3800128 , 0.3998677 , 0.36308518, 0.39958294,
              0.38374275, 0.37822789, 0.37836398, 0.38151351, 0.36924892,
              0.38210833, 0.36464998, 0.38216481, 0.39524334, 0.35991591,
              0.39317798, 0.38769343, 0.38178329, 0.36453202, 0.39463086,
              0.38978188, 0.37299139, 0.38740477, 0.41016846, 0.38995433,
              0.364197  , 0.38840383, 0.38041277, 0.37452238, 0.36725453,
              0.38565668, 0.3637653 , 0.37566207, 0.38089131, 0.36369541,
              0.37689822, 0.37964618, 0.39119645, 0.3921312 , 0.38758545,
              0.37537762, 0.39568623, 0.37041133, 0.39095267, 0.36744416,
      ...
              0.37274202, 0.38693538, 0.38454567, 0.39920365, 0.37956687,
              0.38168832, 0.37162198, 0.37710879, 0.36798413, 0.3936924 ,
              0.37865187, 0.38839493, 0.36901106, 0.39057439, 0.37246345,
              0.38880536, 0.37711224, 0.40336544, 0.38517725, 0.39397022,
              0.37759222, 0.35271155, 0.37873965, 0.36645727, 0.38628891,
              0.38353996, 0.38626757, 0.3719754 , 0.37232457, 0.40274545,
              0.34771228, 0.38651038, 0.38159536, 0.35639062, 0.3633836 ,
              0.38986708, 0.37978639, 0.39185708, 0.36607203, 0.3676719 ,
              0.37168225, 0.38264759, 0.38247754, 0.37833106, 0.39431684,
              0.3914877 , 0.37840656, 0.38997239, 0.39998104, 0.38153429,
              0.37384054, 0.37724475, 0.38367211, 0.36276317, 0.38396537,
              0.37216683, 0.37623201, 0.38823402, 0.37210485, 0.38048027,
              0.37770978, 0.36971562, 0.37398579, 0.36415585, 0.37121288,
              0.3787213 , 0.37648446, 0.39896752, 0.35716992, 0.40190447,
              0.41195481, 0.35760532, 0.36887229, 0.3842352 , 0.38936885,
              0.40469189, 0.33035592, 0.41619787, 0.38499332, 0.37796143,
              0.38123385, 0.36578041, 0.38010795, 0.37893521, 0.37469515,
              0.33789093, 0.39393838, 0.40021458, 0.36334789, 0.38779268,
              0.35800917, 0.3464761 , 0.4029188 , 0.37690717, 0.37493058,
              0.35557172, 0.37269223, 0.34981788, 0.36327578, 0.36212036]])

    • a_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      -0.2317 -0.4188 ... -0.3271 0.5805

      array([[[-0.23171098, -0.41879792, -0.1975896 , ...,  1.68701676,
               -0.08148876,  0.84863651],
              [ 1.21977415,  0.29961867,  0.46438834, ...,  1.75609104,
               -0.22301909,  0.10078001],
              [ 0.8912923 ,  1.23044433,  1.0601216 , ...,  1.90367428,
               -0.94484705,  0.06130805],
              ...,
              [ 0.61985202,  0.52079971,  0.68790904, ..., -0.55019229,
                0.42623162,  1.31008424],
              [ 0.11238105, -0.87751636,  0.8228818 , ...,  0.83581557,
                0.19559232,  0.50563555],
              [ 0.03245844, -0.16189364,  1.07263653, ...,  1.34605525,
               -0.32708694,  0.58054933]]], shape=(1, 1000, 20))

    • scaler_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      0.008074 0.124 ... -0.007553

      array([[[ 0.00807429,  0.12395304, -0.23889308, ...,  0.01208114,
               -0.1462931 , -0.373309  ],
              [-0.07417332,  0.13150385, -0.11710106, ...,  0.04507649,
               -0.06660793, -0.20692631],
              [ 0.02372759,  0.13219046, -0.01887913, ...,  0.07843803,
               -0.13795075, -0.10096361],
              ...,
              [-0.15480335,  0.20844752, -0.00782207, ...,  0.00105329,
               -0.04207431, -0.16912047],
              [-0.09420137,  0.03935843,  0.0125785 , ..., -0.13069304,
               -0.19794295, -0.07011762],
              [-0.06434423,  0.14009702,  0.04477589, ..., -0.05883289,
               -0.0592792 , -0.00755296]]], shape=(1, 1000, 20))

    • rl.alpha_Intercept
      (chain, draw)
      float64
      0.6798 0.6869 ... 0.7553 0.6144

      array([[0.67981451, 0.68686736, 0.68082074, 0.7901338 , 0.59723767,
              0.78260379, 0.57688863, 0.79139111, 0.6637761 , 0.74490533,
              0.84441795, 0.80207425, 0.60019375, 0.8481621 , 0.68522582,
              0.71024341, 0.66620078, 0.68433684, 0.71670737, 0.66328737,
              0.78819137, 0.63560364, 0.87508355, 0.68184612, 0.75426002,
              0.65516793, 0.78055336, 0.61218487, 0.68851134, 0.60475108,
              0.6784454 , 0.63098478, 0.75832838, 0.6649001 , 0.80314143,
              0.80186905, 0.71151081, 0.75428863, 0.68474017, 0.75493681,
              0.80489748, 0.64513994, 0.68157431, 0.72463278, 0.72734395,
              0.67916935, 0.79659412, 0.74244689, 0.79301655, 0.67322377,
              0.66685744, 0.72338557, 0.77725868, 0.71432524, 0.7103513 ,
              0.79667868, 0.65410764, 0.78664728, 0.69308851, 0.82279596,
              0.68986628, 0.68103067, 0.85526019, 0.62039723, 0.77387761,
              0.63352147, 0.76467683, 0.59216205, 0.80968998, 0.67409506,
              0.73088954, 0.76919809, 0.74533741, 0.71609783, 0.66974316,
              0.84001144, 0.64318688, 0.71115672, 0.66212016, 0.72574243,
              0.71272818, 0.71898769, 0.74754889, 0.68193669, 0.80012137,
              0.64005786, 0.6741602 , 0.76564657, 0.76930392, 0.75622214,
              0.74222253, 0.81114309, 0.83619285, 0.61151945, 0.75947429,
              0.67126027, 0.63481295, 0.6959517 , 0.6851581 , 0.82482544,
      ...
              0.68827554, 0.83737369, 0.87204962, 0.68163882, 0.76682429,
              0.65783056, 0.65368253, 0.69043754, 0.82216799, 0.6538156 ,
              0.93932833, 0.90391781, 0.66287597, 0.85534831, 0.83893325,
              0.67309521, 0.75384936, 0.84359486, 0.79273253, 0.73031399,
              0.61239545, 0.81222361, 0.62413585, 0.70724998, 0.63648772,
              0.75173204, 0.7345834 , 0.81831758, 0.72086196, 0.72667022,
              0.65978129, 0.7086867 , 0.71688173, 0.67320289, 0.64837575,
              0.71288695, 0.67323977, 0.75037787, 0.66395608, 0.77510554,
              0.73260492, 0.6391988 , 0.63660074, 0.59685252, 0.8475497 ,
              0.83617637, 0.65506532, 0.71788155, 0.8429329 , 0.76410201,
              0.62105728, 0.63528185, 0.69773924, 0.83157808, 0.67764764,
              0.73283927, 0.68416303, 0.74281402, 0.63772126, 0.83335348,
              0.67256604, 0.66415624, 0.66412155, 0.68714561, 0.76360103,
              0.67999903, 0.68371581, 0.70509652, 0.67579059, 0.73179815,
              0.66667485, 0.80329659, 0.54617839, 0.89184386, 0.80848606,
              0.78835418, 0.68163496, 0.76682457, 0.66981904, 0.68532742,
              0.71204748, 0.59344547, 0.849233  , 0.59440271, 0.83002553,
              0.80473961, 0.70213396, 0.72167669, 0.71155166, 0.71790473,
              0.68828105, 0.73439188, 0.76649404, 0.60220135, 0.82280767,
              0.64489571, 0.73529588, 0.66229025, 0.75534503, 0.61442176]])

    • t_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      -0.3819 -1.056 ... -0.1613 -0.8788

      array([[[-0.38194485, -1.05586571,  0.02628284, ...,  0.49830232,
                1.537318  , -1.08462159],
              [-0.06793551, -0.39318713, -0.8529339 , ..., -0.69095614,
                0.82344507,  0.62733361],
              [ 0.06711075, -0.62281545, -0.80937158, ..., -0.20139458,
                0.77929227,  1.08127646],
              ...,
              [-0.85929822,  0.23799789, -0.13492851, ...,  2.25398645,
                0.97625651,  0.73325622],
              [-0.77171718, -0.19800345, -0.02239689, ...,  0.96131147,
                0.22079336, -0.53635593],
              [-0.62258993,  0.28542593, -0.86663604, ...,  0.20092424,
               -0.1612862 , -0.87878929]]], shape=(1, 1000, 20))

    • rl.alpha_1|participant_id
      (chain, draw, rl.alpha_1|participant_id__factor_dim)
      float64
      -0.0368 -0.002968 ... 0.02581

      array([[[-0.03679909, -0.00296798,  0.02187738, ..., -0.0145418 ,
                0.01633742, -0.04137242],
              [-0.07928371,  0.0038912 ,  0.01967048, ..., -0.00428708,
                0.01794866, -0.02987455],
              [-0.0499396 , -0.0053482 ,  0.01877159, ..., -0.01667856,
                0.01620367, -0.02774294],
              ...,
              [ 0.0325509 , -0.06555012,  0.09426108, ..., -0.04161504,
               -0.03901267, -0.02799446],
              [ 0.02629609,  0.02591385,  0.05631842, ...,  0.06308307,
               -0.08170626, -0.06333946],
              [-0.00738545,  0.00101153,  0.01249355, ...,  0.03750928,
               -0.02891021,  0.02580884]]], shape=(1, 1000, 20))

    • t_1|participant_id_sigma
      (chain, draw)
      float64
      0.005011 0.01054 ... 0.01283

      array([[0.00501128, 0.01054137, 0.01524848, 0.01572233, 0.01123321,
              0.02088035, 0.01831254, 0.00327885, 0.03241795, 0.02746393,
              0.03341502, 0.02660728, 0.01023465, 0.00602655, 0.02133911,
              0.01639495, 0.01641469, 0.00732684, 0.01476469, 0.01018447,
              0.01110854, 0.01133345, 0.0181967 , 0.03814336, 0.01698099,
              0.01707232, 0.02430018, 0.03937436, 0.01254125, 0.01056846,
              0.01432492, 0.00708839, 0.01031127, 0.01530797, 0.02174614,
              0.00465919, 0.01105182, 0.01017195, 0.00896005, 0.00954741,
              0.01119766, 0.00811662, 0.01461603, 0.0127413 , 0.02828874,
              0.02555107, 0.01734767, 0.0190448 , 0.02087055, 0.03583015,
              0.04173709, 0.02971832, 0.03584903, 0.01910053, 0.01942048,
              0.02354357, 0.03057129, 0.01812464, 0.01374013, 0.00973919,
              0.02834382, 0.01809401, 0.01016741, 0.0220994 , 0.02996113,
              0.02165008, 0.01708937, 0.02227103, 0.0179003 , 0.01921443,
              0.06066329, 0.03116757, 0.02708662, 0.03702384, 0.02070866,
              0.0223791 , 0.02895745, 0.03398312, 0.01433851, 0.02544746,
              0.02896384, 0.02104393, 0.0236799 , 0.00513392, 0.0071528 ,
              0.01230246, 0.01748626, 0.01516094, 0.01257643, 0.01030019,
              0.02645552, 0.02005504, 0.00846291, 0.01688451, 0.03032933,
              0.01407554, 0.01482068, 0.02838158, 0.01617578, 0.01436258,
      ...
              0.00769785, 0.00722986, 0.00483249, 0.00793816, 0.02273649,
              0.02143647, 0.01953009, 0.01886915, 0.00800449, 0.03452717,
              0.02255402, 0.00497536, 0.02768511, 0.01233675, 0.0166275 ,
              0.02297582, 0.00931896, 0.01593861, 0.01705365, 0.01792476,
              0.02646566, 0.00765189, 0.00926627, 0.01018181, 0.01377083,
              0.00187705, 0.00207205, 0.01604242, 0.02983713, 0.01253906,
              0.00325928, 0.00345325, 0.00371499, 0.01734627, 0.01096586,
              0.00860599, 0.02413122, 0.02761615, 0.03154432, 0.03780112,
              0.03245063, 0.02163091, 0.02759888, 0.02347867, 0.02736987,
              0.02984709, 0.03566231, 0.03449733, 0.04822428, 0.01044563,
              0.04570791, 0.03929428, 0.02525433, 0.02761916, 0.0190146 ,
              0.0089807 , 0.01070961, 0.00946985, 0.00916451, 0.03357767,
              0.02613565, 0.01686198, 0.02538053, 0.00329155, 0.02211504,
              0.02594151, 0.02770055, 0.013735  , 0.015314  , 0.03002161,
              0.02380949, 0.02196524, 0.01889795, 0.01659494, 0.00675431,
              0.0043088 , 0.00802228, 0.00460835, 0.01817774, 0.02026742,
              0.01911117, 0.03292088, 0.01199471, 0.02061779, 0.01263754,
              0.01009841, 0.01969273, 0.00482563, 0.0040375 , 0.00457878,
              0.03901345, 0.02216891, 0.02262017, 0.00386708, 0.02010124,
              0.02148183, 0.01816215, 0.01979764, 0.02497194, 0.01283101]])

    • a_1|participant_id_sigma
      (chain, draw)
      float64
      0.07174 0.06757 ... 0.0261 0.0246

      array([[0.07173519, 0.06757325, 0.04499785, 0.04636885, 0.01237667,
              0.04267171, 0.05554642, 0.06060978, 0.05215793, 0.02850317,
              0.03103155, 0.04809045, 0.0514551 , 0.0457768 , 0.05866729,
              0.08722717, 0.05298975, 0.04059   , 0.02766446, 0.02265341,
              0.02748525, 0.06488209, 0.06979739, 0.0523025 , 0.04009004,
              0.03088425, 0.04057532, 0.09104736, 0.03253716, 0.08663923,
              0.09066234, 0.07936465, 0.0375405 , 0.05337286, 0.05909928,
              0.05040438, 0.04776153, 0.04988124, 0.0486959 , 0.02952079,
              0.06877796, 0.02551684, 0.03858532, 0.02931617, 0.04900554,
              0.04670761, 0.03373212, 0.02967967, 0.01478621, 0.02988847,
              0.03320885, 0.05277283, 0.02446208, 0.04197646, 0.04747993,
              0.04228589, 0.02749925, 0.05285338, 0.06093665, 0.0511434 ,
              0.02655579, 0.05046508, 0.02910316, 0.03803855, 0.04127443,
              0.04516493, 0.02779485, 0.04700665, 0.07164582, 0.08754618,
              0.0449107 , 0.02990185, 0.05205801, 0.05596508, 0.07221751,
              0.05635773, 0.04838369, 0.04814239, 0.04486825, 0.05520174,
              0.04344116, 0.03654437, 0.02366813, 0.06180248, 0.03462425,
              0.04867306, 0.06028767, 0.06267276, 0.03060344, 0.05718269,
              0.03798494, 0.05887488, 0.04177753, 0.05378804, 0.0655356 ,
              0.01197282, 0.02412687, 0.03827697, 0.02230057, 0.01342497,
      ...
              0.07108901, 0.06264612, 0.05273294, 0.05212277, 0.04830978,
              0.04682725, 0.04120849, 0.07315502, 0.06035002, 0.07576027,
              0.03806807, 0.04511131, 0.02223927, 0.08206318, 0.0638094 ,
              0.04557301, 0.05912782, 0.05119268, 0.072299  , 0.07378679,
              0.04073594, 0.05577519, 0.02791974, 0.06947737, 0.03069103,
              0.05158493, 0.02453049, 0.05007425, 0.0485951 , 0.04382479,
              0.06784222, 0.03155337, 0.03299433, 0.04583786, 0.05865193,
              0.0570478 , 0.05595955, 0.02895182, 0.03059495, 0.02830817,
              0.07088474, 0.07970198, 0.0797958 , 0.06371377, 0.06982702,
              0.06716872, 0.06892839, 0.04619751, 0.06248188, 0.0696298 ,
              0.04804402, 0.08522746, 0.04549831, 0.0624039 , 0.03833435,
              0.0506618 , 0.0578271 , 0.04922972, 0.02713636, 0.04292028,
              0.05746022, 0.09334869, 0.04619903, 0.04758745, 0.05265474,
              0.08781584, 0.09474239, 0.0687736 , 0.03655969, 0.04010972,
              0.03397715, 0.03441265, 0.06821884, 0.06789032, 0.05941363,
              0.0499974 , 0.04632217, 0.05074637, 0.04289316, 0.05860868,
              0.05266718, 0.04189182, 0.11079897, 0.0381235 , 0.08006359,
              0.04836054, 0.04960625, 0.05342488, 0.04216447, 0.05301546,
              0.027539  , 0.03143972, 0.04974105, 0.0419891 , 0.07074808,
              0.05718809, 0.03930726, 0.0410844 , 0.02610259, 0.02460054]])

    • rl.alpha_neg_Intercept
      (chain, draw)
      float64
      0.6904 0.7299 ... 0.7092 0.6561

      array([[0.69038683, 0.72993853, 0.7449339 , 0.64959241, 0.7358063 ,
              0.56610348, 0.63400391, 0.70712486, 0.70312414, 0.58867178,
              0.60437564, 0.56343472, 0.80628413, 0.5610069 , 0.84320671,
              0.60689762, 0.79474184, 0.75285952, 0.78956679, 0.78258027,
              0.65821826, 0.74786894, 0.58981609, 0.81946753, 0.64302529,
              0.6890724 , 0.70809536, 0.66629117, 0.72622853, 0.74892018,
              0.78987283, 0.80853928, 0.55621314, 0.70445503, 0.65728093,
              0.63167132, 0.79657891, 0.57724362, 0.64194875, 0.66383771,
              0.65373719, 0.72209192, 0.72280303, 0.69331169, 0.63539943,
              0.66447656, 0.69556475, 0.67674146, 0.60861006, 0.70831231,
              0.69182503, 0.7437137 , 0.67754572, 0.68430712, 0.77185816,
              0.69540557, 0.6566537 , 0.70667704, 0.65619659, 0.71714523,
              0.65192555, 0.68259091, 0.67759049, 0.68994186, 0.67629339,
              0.69068016, 0.64254273, 0.81895804, 0.63204506, 0.71560487,
              0.66035689, 0.65991725, 0.71963277, 0.67202727, 0.64756835,
              0.66592885, 0.66469185, 0.59403817, 0.74619752, 0.72349888,
              0.6434175 , 0.69099596, 0.59321089, 0.73872918, 0.60717964,
              0.80676915, 0.6321594 , 0.66167166, 0.70488998, 0.64940243,
              0.77911958, 0.58352784, 0.62645531, 0.76020457, 0.71271209,
              0.62371312, 0.7181133 , 0.6306151 , 0.68634513, 0.57125826,
      ...
              0.6778626 , 0.64462731, 0.77318772, 0.62675624, 0.70165054,
              0.6794069 , 0.71044425, 0.69523635, 0.68801706, 0.63134453,
              0.67810832, 0.58096296, 0.81764392, 0.62740541, 0.67243539,
              0.69584001, 0.71009613, 0.58610207, 0.76055133, 0.68040318,
              0.75626844, 0.53410677, 0.71096767, 0.70722735, 0.68572318,
              0.6858935 , 0.64891125, 0.76460494, 0.6334908 , 0.66689878,
              0.69789812, 0.75922621, 0.65487747, 0.73391358, 0.69597517,
              0.68389748, 0.68557007, 0.65230061, 0.68104211, 0.74261849,
              0.67051668, 0.76690238, 0.75064105, 0.85506839, 0.49090012,
              0.50390318, 0.83561221, 0.76189344, 0.54945598, 0.88430624,
              0.9261316 , 0.9287266 , 0.86586296, 0.53418688, 0.66337421,
              0.66773149, 0.72422049, 0.66359905, 0.6138246 , 0.70289211,
              0.7862396 , 0.75785777, 0.70385503, 0.68031133, 0.66739917,
              0.72694522, 0.72650672, 0.65807471, 0.79952224, 0.69639962,
              0.64432701, 0.70862006, 0.6959433 , 0.61072779, 0.65562685,
              0.69047368, 0.73734077, 0.65500192, 0.78178878, 0.76598014,
              0.74271528, 0.7199929 , 0.61413797, 0.68290769, 0.64092818,
              0.66921406, 0.6663995 , 0.66369316, 0.73950862, 0.66519456,
              0.69377535, 0.69213989, 0.61052153, 0.81529134, 0.58891638,
              0.69488287, 0.7097902 , 0.69254868, 0.70920397, 0.65613717]])

    • rl.alpha_1|participant_id_offset
      (chain, draw, rl.alpha_1|participant_id__factor_dim)
      float64
      -0.993 -0.08009 ... -1.002 0.8942

      array([[[-0.99297873, -0.08008733,  0.5903345 , ..., -0.39239292,
                0.44084537, -1.11638438],
              [-2.25666873,  0.11075601,  0.55988493, ..., -0.12202411,
                0.51087658, -0.85032562],
              [-1.72458347, -0.18469131,  0.64824644, ..., -0.57596713,
                0.55956764, -0.95805761],
              ...,
              [ 0.38586717, -0.77704876,  1.11739629, ..., -0.49331586,
               -0.46246674, -0.33185388],
              [ 0.4032201 ,  0.39735886,  0.86357775, ...,  0.96730587,
               -1.25287092, -0.97123734],
              [-0.25588914,  0.03504735,  0.43287293, ...,  1.29961124,
               -1.00167305,  0.89421751]]], shape=(1, 1000, 20))

    • t_1|participant_id_mu
      (chain, draw)
      float64
      0.01511 -0.2057 ... 0.3788 0.2383

      array([[ 0.01511467, -0.20572407, -0.12132609,  0.29957555, -0.36195211,
               0.08692273, -0.04953266,  0.25743939, -0.25711457,  0.13201344,
              -0.10551054, -0.08230923, -0.2923595 ,  0.32416111, -0.1016563 ,
               0.13287597,  0.43676477,  0.04412088,  0.39005455,  0.10163718,
              -0.08182372, -0.05548793,  0.25117531, -0.12116738,  0.24189656,
              -0.165815  , -0.33789161, -0.03977139,  0.29917878, -0.63894304,
              -0.11424422, -0.64521075,  0.16773168, -0.20398306, -0.07798314,
              -0.12613954,  0.16124638,  0.48045142, -0.3773307 ,  0.37257459,
              -0.42499062,  0.26620606,  0.10669677, -0.14281455,  0.19341479,
               0.33253584, -0.32607866,  0.33052039,  0.08143363, -0.05091278,
               0.03071056, -0.45228429, -0.2243013 ,  0.11463483,  0.14855446,
              -0.31041119,  0.30208406, -0.30463904,  0.29198677, -0.12261104,
               0.02274629, -0.0068877 , -0.09590145, -0.5335171 ,  0.52055268,
              -0.48551565,  0.25294147, -0.31554745,  0.37304265,  0.06846637,
              -0.08281051,  0.20657642, -0.2660431 ,  0.13790166, -0.12815107,
              -0.31911208, -0.31785483,  0.3160004 , -0.33539657,  0.07793024,
              -0.00317632, -0.02126081, -0.06100694,  0.05115548, -0.02259676,
               0.18874049, -0.31440947, -0.44099992, -0.52542371,  0.32414841,
              -0.29373919,  0.15891968, -0.03847389,  0.06819933, -0.3513549 ,
               0.46970234, -0.04929152,  0.0364425 , -0.02323526,  0.04323053,
      ...
              -0.23086025, -0.10247109,  0.01207696, -0.02682633,  0.07742928,
              -0.08884781,  0.06236189,  0.17944289, -0.25673309,  0.28788289,
              -0.32646717,  0.34280961, -0.37891675,  0.43436695,  0.51777332,
              -0.43455461,  0.24662785, -0.31914911,  0.16405783, -0.04071189,
               0.12011175,  0.13449565,  0.0397748 , -0.19722587,  0.46897564,
              -0.44703548, -0.23853413,  0.12142134,  0.00669417, -0.13471392,
               0.11397265, -0.14171136,  0.23128641,  0.08774496,  0.14052393,
              -0.47310229, -0.04329747, -0.09637734,  0.12819152, -0.35142379,
              -0.22245996,  0.26530796,  0.20436492, -0.16140517,  0.06555417,
               0.05725287, -0.09859644, -0.01573384,  0.2504415 , -0.26193004,
              -0.05368492,  0.17603271, -0.02226524,  0.11519561,  0.10856586,
               0.04846163,  0.29524464, -0.02499308,  0.05609767, -0.06856198,
               0.10040024, -0.08345717,  0.01023871,  0.10211745, -0.08120329,
               0.15764293,  0.12175832, -0.32749953,  0.33365644, -0.24620413,
              -0.17989596,  0.27716284,  0.42998776, -0.54015241, -0.26307347,
              -0.04092977,  0.11086023, -0.12574464,  0.38986765, -0.35788988,
              -0.21738949, -0.37756401,  0.4601979 , -0.45965829,  0.4714485 ,
              -0.32598537, -0.01618817,  0.08385242,  0.06245422, -0.11120679,
               0.09760044,  0.45732136,  0.20062178, -0.21334609,  0.16994305,
              -0.21298984,  0.21129826,  0.01205766,  0.37883855,  0.23826645]])

    • z_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      1.153 -1.022 ... -0.6438 -1.911

      array([[[ 1.15270492, -1.0217935 ,  0.88096742, ...,  1.1647648 ,
                0.01443685,  1.86731971],
              [ 1.25122774,  0.16012276, -0.1706174 , ...,  0.29988808,
               -0.25394959,  1.19874627],
              [ 0.11431206, -0.06624679, -0.66381407, ...,  0.31994262,
               -0.25568053,  1.76222547],
              ...,
              [-0.59279102, -1.15911924, -1.84826571, ...,  0.11254186,
               -1.10042359, -0.952858  ],
              [-2.06814001, -1.28752815, -1.1797962 , ...,  0.21728314,
                0.03660396, -1.43134713],
              [-0.2176119 ,  0.5409327 , -0.4124358 , ..., -0.77447715,
               -0.64377527, -1.91069582]]], shape=(1, 1000, 20))

    • z_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      0.005671 -0.005027 ... -0.04568

      array([[[ 5.67118581e-03, -5.02711552e-03,  4.33426616e-03, ...,
                5.73051911e-03,  7.10277927e-05,  9.18701467e-03],
              [ 7.24531345e-03,  9.27200993e-04, -9.87970832e-04, ...,
                1.73652089e-03, -1.47051120e-03,  6.94141621e-03],
              [ 8.50950493e-04, -4.93147731e-04, -4.94149862e-03, ...,
                2.38168502e-03, -1.90331155e-03,  1.31181835e-02],
              ...,
              [-9.67131312e-03, -1.89108890e-02, -3.01542297e-02, ...,
                1.83610679e-03, -1.79532767e-02, -1.55457621e-02],
              [-1.60776395e-02, -1.00091935e-02, -9.17168949e-03, ...,
                1.68915061e-03,  2.84557715e-04, -1.11272366e-02],
              [-5.20286564e-03,  1.29331170e-02, -9.86089483e-03, ...,
               -1.85169126e-02, -1.53919716e-02, -4.56826744e-02]]],
            shape=(1, 1000, 20))

    • scaler_1|participant_id_sigma
      (chain, draw)
      float64
      0.1388 0.137 ... 0.2674 0.09901

      array([[0.13883402, 0.13695573, 0.11849868, 0.2370089 , 0.14949371,
              0.15389322, 0.13428827, 0.12770186, 0.13803292, 0.11683156,
              0.1172641 , 0.13595729, 0.10990215, 0.11187867, 0.09951282,
              0.10291779, 0.10985967, 0.13976347, 0.1391927 , 0.13752252,
              0.18606117, 0.10043801, 0.07932981, 0.02230673, 0.04396884,
              0.12340407, 0.19749686, 0.08094822, 0.03136297, 0.00714891,
              0.0065612 , 0.04850566, 0.12790267, 0.12973162, 0.07631307,
              0.15333151, 0.15028776, 0.15365694, 0.01957529, 0.11991931,
              0.04479693, 0.01791469, 0.06115986, 0.17033434, 0.06145518,
              0.15351383, 0.17869024, 0.21799441, 0.15775311, 0.21986665,
              0.11163786, 0.08950804, 0.08507637, 0.03602241, 0.14550513,
              0.08859383, 0.12975492, 0.07733683, 0.12455398, 0.15202855,
              0.13445165, 0.08721999, 0.14038856, 0.06489532, 0.05724345,
              0.17623275, 0.13648628, 0.1160806 , 0.17573855, 0.10768908,
              0.06717272, 0.18058678, 0.13521496, 0.17168171, 0.1304983 ,
              0.09483273, 0.11210288, 0.09765789, 0.12329164, 0.12678427,
              0.14670532, 0.13399923, 0.08466465, 0.10880114, 0.10123768,
              0.17289792, 0.0464579 , 0.15901177, 0.10550302, 0.09532858,
              0.09053835, 0.06695293, 0.06811207, 0.01233338, 0.12784902,
              0.12446723, 0.11835842, 0.12501622, 0.09850538, 0.02374594,
      ...
              0.09596475, 0.1367606 , 0.18861998, 0.11180634, 0.06732487,
              0.10590262, 0.08391199, 0.0614631 , 0.10983597, 0.05508622,
              0.06829853, 0.09510736, 0.112354  , 0.07264883, 0.04303162,
              0.02161306, 0.07118339, 0.04347767, 0.03558323, 0.0258348 ,
              0.09562234, 0.17696993, 0.14855759, 0.19776204, 0.17646268,
              0.22519951, 0.08488752, 0.02298579, 0.06287322, 0.19317189,
              0.16866511, 0.06259875, 0.10964955, 0.10266083, 0.11543212,
              0.08952323, 0.15077565, 0.01614116, 0.02593329, 0.06664143,
              0.0457872 , 0.02988396, 0.02502704, 0.03138401, 0.03372625,
              0.03236418, 0.01734978, 0.03269736, 0.10073787, 0.03431711,
              0.05601549, 0.01738471, 0.01977466, 0.05514682, 0.14093553,
              0.12028046, 0.11640691, 0.10818165, 0.10059389, 0.11934271,
              0.18416174, 0.17706813, 0.04270399, 0.13361721, 0.08517431,
              0.11690971, 0.12934158, 0.15462919, 0.06827055, 0.08945355,
              0.09625059, 0.1678879 , 0.21573586, 0.11333648, 0.08457554,
              0.09229381, 0.11676618, 0.04847994, 0.066074  , 0.06814324,
              0.04756327, 0.04610921, 0.05970327, 0.05357878, 0.10954619,
              0.09958176, 0.01296234, 0.02367471, 0.03755727, 0.0652909 ,
              0.03995674, 0.07921431, 0.09379009, 0.14253633, 0.11984931,
              0.13391687, 0.29358021, 0.13805487, 0.26735115, 0.09900586]])

    • rl.alpha_neg_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      0.914 -0.4167 1.184 ... 1.136 2.077

      array([[[ 0.91395466, -0.4167473 ,  1.18360938, ..., -0.64665616,
                0.11915031,  1.04947734],
              [ 1.12738671, -0.58821763, -1.12966871, ..., -0.04491387,
                0.37136067,  0.08228087],
              [ 0.0107396 , -0.47477346, -1.01323521, ...,  0.03814794,
                0.78954875,  0.78061348],
              ...,
              [ 0.47141322, -0.68800843, -0.4876386 , ..., -1.09840104,
                0.52757893,  0.91085337],
              [ 1.48082587, -0.59103129, -0.73255405, ...,  1.02278363,
                0.87348162,  0.71020961],
              [ 0.46571013,  0.54413125, -0.92527178, ...,  1.25565021,
                1.13607474,  2.07711204]]], shape=(1, 1000, 20))

    • t_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      -0.001914 -0.005291 ... -0.01128

      array([[[-0.00191403, -0.00529124,  0.00013171, ...,  0.00249713,
                0.00770393, -0.00543534],
              [-0.00071613, -0.00414473, -0.00899109, ..., -0.00728362,
                0.00868024,  0.00661295],
              [ 0.00102334, -0.00949699, -0.01234169, ..., -0.00307096,
                0.01188302,  0.01648782],
              ...,
              [-0.01701208,  0.0047118 , -0.00267127, ...,  0.04462361,
                0.01932757,  0.01451674],
              [-0.01927128, -0.00494453, -0.00055929, ...,  0.02400581,
                0.00551364, -0.01339385],
              [-0.00798846,  0.0036623 , -0.01111982, ...,  0.00257806,
               -0.00206947, -0.01127576]]], shape=(1, 1000, 20))

    • rl.alpha_neg_1|participant_id_mu
      (chain, draw)
      float64
      -0.1912 0.1659 ... 0.2265 -0.254

      array([[-1.91163480e-01,  1.65893333e-01, -2.41744087e-01,
               1.27074881e-02,  8.65219394e-02, -1.68640810e-01,
              -2.80162179e-01,  2.48999455e-01, -2.53842287e-01,
              -2.78177931e-02,  7.82455227e-02,  1.37383617e-01,
               1.02398968e-01, -1.35284889e-01,  9.71249071e-02,
              -1.61088444e-01,  4.34891944e-02, -1.68586124e-01,
              -3.64298683e-01,  1.30713495e-01, -1.52538122e-01,
               1.29101166e-01, -1.84451937e-01, -9.33884279e-03,
              -1.35247569e-01, -2.01169718e-01, -4.90062036e-01,
               2.27070775e-01, -2.49072008e-02, -4.63648471e-01,
              -5.41428038e-01,  2.33896144e-01, -2.61038191e-01,
               5.11841845e-01, -3.44227470e-01,  2.79644825e-01,
              -2.59579326e-01,  4.18372793e-01, -4.27188476e-01,
               4.40970589e-01, -2.44833096e-01,  1.76681010e-01,
              -1.50931194e-01,  3.93936977e-02, -4.17818146e-02,
              -1.39548354e-01, -2.97959458e-01,  1.60627197e-01,
              -1.92047887e-01,  1.45341660e-01, -6.31980763e-02,
               5.20091913e-01,  7.69077395e-02, -1.42362001e-01,
               4.04356217e-02,  1.21898531e-02, -2.30951203e-01,
               9.68739609e-02, -1.04688897e-01,  2.08765613e-01,
      ...
              -1.00663932e-01,  1.25416575e-01, -3.28134887e-02,
              -8.43287247e-02,  2.46257734e-02,  1.78017922e-01,
               2.97961190e-01, -3.92148217e-01, -1.82328010e-01,
              -4.84184010e-02, -8.50307090e-02,  1.34247380e-01,
              -1.62616159e-01,  3.34091534e-01, -1.01267376e-01,
              -6.01855650e-02,  3.69622343e-02, -1.01871409e-02,
               2.67536657e-01, -2.86256618e-01,  2.46299046e-01,
              -2.56669684e-01,  1.30927446e-01, -2.22288078e-01,
              -2.47523962e-01, -2.99566265e-01,  3.32075591e-01,
              -9.51815525e-02, -2.50111982e-01,  8.20012610e-02,
               5.18387265e-02,  1.19466233e-02,  2.52066309e-01,
               2.20709044e-01, -2.01003141e-01,  1.96076467e-01,
               4.18712536e-01, -4.57571314e-01, -4.40509016e-01,
              -4.43448800e-01,  4.28154330e-01, -4.41594413e-01,
               3.93838771e-01, -3.44465968e-01,  1.26766927e-01,
               1.31476871e-01, -1.40187871e-01, -1.05196070e-02,
               6.78692385e-02,  1.33894131e-01,  4.15525201e-01,
              -6.88482234e-02,  4.93912388e-02,  1.16871128e-01,
              -9.04504771e-02, -1.78027319e-02,  2.26549246e-01,
              -2.53989099e-01]])

    • t_Intercept
      (chain, draw)
      float64
      0.428 0.4182 ... 0.413 0.4232

      array([[0.42795654, 0.41823627, 0.42314501, 0.41735082, 0.41619463,
              0.4071601 , 0.44056347, 0.40919825, 0.44195089, 0.40883328,
              0.41523489, 0.4073094 , 0.41784601, 0.41449605, 0.41930529,
              0.41235779, 0.41161047, 0.42033639, 0.41774312, 0.41379587,
              0.41383922, 0.40693655, 0.42955936, 0.41418968, 0.41254351,
              0.42924322, 0.42080165, 0.43080557, 0.43605201, 0.41229373,
              0.42057533, 0.42022927, 0.41618287, 0.42128428, 0.41670349,
              0.42299527, 0.41732937, 0.42164025, 0.41553528, 0.42166416,
              0.41137296, 0.42285852, 0.40846821, 0.41484647, 0.42059521,
              0.42133152, 0.42158073, 0.42040357, 0.42984636, 0.42743805,
              0.42278639, 0.42179975, 0.41363187, 0.42318039, 0.42539817,
              0.4297697 , 0.4235959 , 0.41558856, 0.43636193, 0.4108576 ,
              0.43260185, 0.42834442, 0.42367378, 0.41962605, 0.43243334,
              0.42984584, 0.42561359, 0.41852482, 0.41641895, 0.43145826,
              0.40381442, 0.41479119, 0.41702356, 0.43002545, 0.42010149,
              0.41170148, 0.43302701, 0.39791374, 0.42541215, 0.40845445,
              0.42475907, 0.43404504, 0.41883115, 0.41076757, 0.42565302,
              0.40129423, 0.41524857, 0.40723944, 0.42275631, 0.42901171,
              0.39332584, 0.43940802, 0.43148146, 0.41921352, 0.41008698,
              0.4226552 , 0.4125507 , 0.43822428, 0.4175532 , 0.42676599,
      ...
              0.41480516, 0.41197798, 0.43395988, 0.41586865, 0.42108214,
              0.42042501, 0.4267316 , 0.42136331, 0.42299723, 0.42363132,
              0.41824481, 0.42352098, 0.42396242, 0.42704201, 0.43566468,
              0.40643914, 0.43827992, 0.41769007, 0.4370298 , 0.41055008,
              0.41835593, 0.41902956, 0.42407741, 0.43170136, 0.40590685,
              0.42157009, 0.40587898, 0.42127369, 0.41705206, 0.42779208,
              0.41438404, 0.42013267, 0.41157184, 0.42234159, 0.42644471,
              0.40914155, 0.42900852, 0.41427944, 0.41756009, 0.41107444,
              0.40630463, 0.43954635, 0.43817099, 0.43187778, 0.41244353,
              0.41295888, 0.41980498, 0.40478264, 0.43506238, 0.41738462,
              0.42641597, 0.44004422, 0.43103076, 0.42204631, 0.42046083,
              0.41429288, 0.41158925, 0.41381992, 0.40917449, 0.43091907,
              0.43580473, 0.42347087, 0.42031143, 0.43045823, 0.41568809,
              0.4309304 , 0.43056802, 0.41485988, 0.41291569, 0.41557259,
              0.43375865, 0.43006285, 0.42962073, 0.42424283, 0.41040709,
              0.42301603, 0.42372394, 0.42292936, 0.40514352, 0.42786558,
              0.43132346, 0.43730487, 0.40677776, 0.4376037 , 0.40862596,
              0.43660138, 0.40711775, 0.41325803, 0.4133131 , 0.40825353,
              0.41293523, 0.4140093 , 0.43740441, 0.41520185, 0.42968083,
              0.4230516 , 0.42253972, 0.42738669, 0.41296092, 0.42321183]])

    • z_1|participant_id_mu
      (chain, draw)
      float64
      -0.3845 -0.2661 ... 0.03367 0.3078

      array([[-0.38445513, -0.26610365, -0.08498686, -0.00518097,  0.08367884,
              -0.29467574,  0.35431146, -0.14396236,  0.10381403,  0.21667349,
              -0.17901817, -0.33327185,  0.24754052, -0.25919081,  0.2637896 ,
              -0.14456564, -0.05142755, -0.28325005,  0.02739128,  0.09912178,
              -0.09467615,  0.08298222, -0.09712408,  0.03332864, -0.07969637,
               0.29089948,  0.13634389,  0.15101199, -0.13260154,  0.06449909,
              -0.00387945, -0.25134126, -0.16414408,  0.03717098, -0.09373394,
               0.07116787, -0.13867478,  0.12306532, -0.07585001,  0.06690887,
               0.22945518,  0.03724327, -0.06931435, -0.28061768, -0.05765187,
              -0.37273685,  0.58480359, -0.6801243 , -0.27759224,  0.20521889,
              -0.22289734, -0.01145584,  0.15782034, -0.13648454,  0.36663022,
               0.17724395,  0.08639322, -0.08428371,  0.04685814, -0.15394183,
               0.17809488, -0.08092882,  0.24664464,  0.27328059, -0.2622707 ,
               0.34573446,  0.14108368, -0.14403198,  0.16835908, -0.32129244,
               0.31566407, -0.13657995,  0.07744273, -0.20195604,  0.24918578,
               0.17200137, -0.22721794,  0.22269238, -0.20577418,  0.38482834,
              -0.51282137,  0.47796978, -0.62885922,  0.64295301, -0.57610829,
               0.46871674, -0.49890055, -0.47671215,  0.31561297, -0.33683737,
               0.40052802, -0.19045125,  0.05500176, -0.05023526, -0.15322155,
              -0.01350676,  0.22827694, -0.07612768,  0.17279737, -0.16783627,
      ...
               0.05810804,  0.14673046,  0.16092736, -0.12904012,  0.15562485,
               0.10650882, -0.09062207, -0.25187423,  0.21226772, -0.14073981,
               0.02111361, -0.08299419, -0.085154  ,  0.13812413, -0.30986806,
               0.37500875, -0.2912714 ,  0.38541369, -0.05595662,  0.06197875,
              -0.13128986, -0.02865051, -0.12845949, -0.28057316,  0.14035149,
              -0.08529648,  0.12743635,  0.09571021, -0.47402108,  0.27496687,
              -0.3983348 ,  0.26769788, -0.32741872, -0.1633967 ,  0.02167076,
              -0.21747759,  0.30507532, -0.26869995, -0.01232208,  0.06016842,
               0.07454836, -0.10253738, -0.13306231,  0.0198188 ,  0.07312948,
               0.13930926, -0.05575929,  0.39240501, -0.20304019,  0.28963143,
               0.50119251, -0.60446654,  0.68061968, -0.45383777, -0.21997373,
               0.23366197,  0.26016504, -0.42278481, -0.12864326,  0.13376956,
              -0.04380211,  0.00986593, -0.0798749 ,  0.39127279, -0.50679482,
               0.39849345,  0.37714201, -0.00898171,  0.01024528,  0.00199428,
               0.40421152, -0.3326053 , -0.21566443,  0.30176821,  0.13736922,
               0.36878817, -0.31998447,  0.34030536, -0.26669462,  0.29168813,
               0.26956602, -0.15026433,  0.11011255, -0.11005266,  0.1113702 ,
               0.33204116, -0.2302861 , -0.04569952, -0.13321691,  0.13976592,
              -0.17235007, -0.12974042, -0.02932819,  0.07935601, -0.15087585,
               0.18029748,  0.08395692,  0.06804948,  0.03367068,  0.30779514]])

    • theta_1|participant_id_offset
      (chain, draw, participant_id__factor_dim)
      float64
      1.081 -0.05683 ... 0.6488 1.496

      array([[[ 1.08064786, -0.05683023,  0.76743687, ..., -0.44403018,
                0.08708229,  2.03109539],
              [ 0.12593339,  0.72860755,  0.53089381, ...,  0.24894858,
               -0.08017794,  0.9067292 ],
              [ 0.27675089,  0.05562065,  0.1021461 , ..., -0.18876699,
               -0.01249637,  1.2954936 ],
              ...,
              [ 0.86379366,  1.01239982,  0.25856898, ..., -1.03481219,
                1.3934954 ,  1.63520832],
              [ 0.24106893,  1.12106309,  1.67942768, ..., -1.10487845,
                0.51805745,  1.14407622],
              [ 0.47765069,  0.80921827,  0.40942822, ..., -1.28309606,
                0.64877481,  1.49613842]]], shape=(1, 1000, 20))

    • z_Intercept
      (chain, draw)
      float64
      0.2854 0.2993 ... 0.3059 0.3075

      array([[0.28535394, 0.2993119 , 0.29992442, 0.30001705, 0.29764792,
              0.29515565, 0.32325465, 0.2878073 , 0.30882159, 0.29431201,
              0.28591197, 0.29060289, 0.28452536, 0.30201674, 0.29842536,
              0.30138836, 0.30396643, 0.29825543, 0.29970298, 0.29891266,
              0.29848888, 0.29731889, 0.30826   , 0.29340137, 0.29532247,
              0.30003169, 0.28520468, 0.30490132, 0.30658569, 0.2966049 ,
              0.29902996, 0.29392082, 0.30393868, 0.29888099, 0.29781481,
              0.29616385, 0.29591465, 0.30844128, 0.28781459, 0.30637182,
              0.28915415, 0.29135074, 0.30371296, 0.29048833, 0.30470957,
              0.29662229, 0.30691191, 0.2880616 , 0.29786101, 0.29779486,
              0.29230968, 0.29317333, 0.28469788, 0.30813368, 0.28128241,
              0.31180583, 0.30222493, 0.29981952, 0.29700221, 0.29289386,
              0.2988997 , 0.30216948, 0.29083848, 0.29883806, 0.30178366,
              0.2991145 , 0.30112754, 0.30014864, 0.30449497, 0.30978842,
              0.29766392, 0.30096508, 0.28718927, 0.31056829, 0.29101269,
              0.28741402, 0.30371914, 0.29243218, 0.30541333, 0.29106231,
              0.30455728, 0.30463482, 0.31312318, 0.29349101, 0.30416171,
              0.28437617, 0.30061386, 0.30475034, 0.2964082 , 0.29993091,
              0.29909109, 0.30222035, 0.30274791, 0.29261432, 0.29820533,
              0.29338119, 0.29479924, 0.3029484 , 0.29626621, 0.30255974,
      ...
              0.29141572, 0.29639209, 0.29098178, 0.30006698, 0.29971707,
              0.29950705, 0.29020791, 0.2851421 , 0.30750325, 0.289773  ,
              0.28760161, 0.30542147, 0.28737187, 0.30796041, 0.31811267,
              0.2921845 , 0.30754245, 0.3023156 , 0.30447257, 0.28723449,
              0.31161573, 0.29243505, 0.30401397, 0.30456455, 0.28960672,
              0.29368438, 0.29711475, 0.30651598, 0.28622175, 0.31339755,
              0.29557761, 0.29164988, 0.29221288, 0.30202907, 0.30065386,
              0.29175901, 0.309105  , 0.29398295, 0.28869259, 0.30419119,
              0.3017563 , 0.31125974, 0.31228975, 0.30649672, 0.30566988,
              0.30437683, 0.29045334, 0.28716103, 0.3090097 , 0.30800029,
              0.31872959, 0.2925329 , 0.30686818, 0.2955717 , 0.29679539,
              0.28793236, 0.29631139, 0.30045706, 0.28527457, 0.31150355,
              0.30507236, 0.30323466, 0.29785615, 0.30323594, 0.29872305,
              0.29477875, 0.29388994, 0.30238001, 0.29453713, 0.29479562,
              0.30652257, 0.29197701, 0.2992591 , 0.29440763, 0.29184112,
              0.30621642, 0.29106598, 0.30719122, 0.29184642, 0.30246756,
              0.30233196, 0.30047371, 0.29551409, 0.29106717, 0.30003597,
              0.30828206, 0.28910783, 0.29019337, 0.29281793, 0.29934729,
              0.30608585, 0.2897666 , 0.31509098, 0.29135533, 0.30471286,
              0.30177028, 0.30854231, 0.31383581, 0.30588341, 0.30745167]])

    • theta_1|participant_id_sigma
      (chain, draw)
      float64
      0.04096 0.05766 ... 0.03672 0.05072

      array([[0.04095581, 0.05766162, 0.0456981 , 0.0449229 , 0.05680918,
              0.0285089 , 0.03453046, 0.02925858, 0.01972195, 0.03455175,
              0.05309737, 0.07252292, 0.06408525, 0.0457302 , 0.0264558 ,
              0.00906853, 0.03019866, 0.05273966, 0.04607154, 0.04347917,
              0.04645139, 0.01041346, 0.01957336, 0.04786794, 0.0426248 ,
              0.03315946, 0.03757007, 0.02729861, 0.0260703 , 0.02594669,
              0.01898064, 0.05302707, 0.04032139, 0.04312885, 0.04127796,
              0.03611918, 0.04183559, 0.04930705, 0.05478179, 0.03391523,
              0.04974164, 0.04620655, 0.03724837, 0.04137495, 0.02041434,
              0.05200859, 0.03244668, 0.0282329 , 0.02945521, 0.0451452 ,
              0.02539614, 0.03684225, 0.04653749, 0.0450934 , 0.06913856,
              0.02417119, 0.03585186, 0.04317742, 0.02577326, 0.02779946,
              0.03247738, 0.04533804, 0.03541986, 0.0360971 , 0.0238885 ,
              0.02069438, 0.038878  , 0.02885935, 0.03182943, 0.02291509,
              0.02891716, 0.02871782, 0.00673819, 0.01307651, 0.02384775,
              0.02994273, 0.05565937, 0.03494539, 0.04928943, 0.05682162,
              0.025282  , 0.04089624, 0.01782405, 0.02373336, 0.05024414,
              0.03824396, 0.02558914, 0.02633754, 0.01215822, 0.01918326,
              0.03405103, 0.06870833, 0.04053351, 0.03664117, 0.04853689,
              0.04432756, 0.05496077, 0.01740131, 0.03378064, 0.05563747,
      ...
              0.03857755, 0.03296677, 0.03408886, 0.03318723, 0.03447868,
              0.02319802, 0.02812294, 0.03807692, 0.03439944, 0.03135831,
              0.02885402, 0.02957312, 0.02448075, 0.02967032, 0.01871197,
              0.02549132, 0.03405451, 0.02321704, 0.05097397, 0.05686479,
              0.03146744, 0.02395055, 0.03277589, 0.05182318, 0.0290177 ,
              0.04424462, 0.02849435, 0.03148975, 0.02766244, 0.02010505,
              0.05530256, 0.03575094, 0.05767258, 0.05120221, 0.07605192,
              0.02817025, 0.04743378, 0.02479317, 0.03139863, 0.0199533 ,
              0.02007286, 0.02464639, 0.02756504, 0.02932353, 0.03871003,
              0.03595645, 0.0533503 , 0.04400041, 0.05284094, 0.06265968,
              0.02906867, 0.04763049, 0.02675166, 0.03843379, 0.0316083 ,
              0.04588487, 0.04134572, 0.03121208, 0.0698196 , 0.02165766,
              0.01876386, 0.00709326, 0.00693389, 0.01086685, 0.01474221,
              0.03171121, 0.02995716, 0.03258617, 0.04536167, 0.03933128,
              0.04802055, 0.035161  , 0.0219788 , 0.02536217, 0.02379602,
              0.03309731, 0.05466112, 0.04622765, 0.05156264, 0.03854952,
              0.03937764, 0.03086262, 0.02307055, 0.05259533, 0.0628386 ,
              0.04608284, 0.02803261, 0.02559631, 0.04605795, 0.00382283,
              0.03184208, 0.04673074, 0.07552445, 0.03476246, 0.03218705,
              0.04577025, 0.04780782, 0.05233627, 0.0367238 , 0.05071736]])

    • theta_1|participant_id_mu
      (chain, draw)
      float64
      -0.2717 -0.5113 ... 0.093 0.1671

      array([[-2.71732086e-01, -5.11296059e-01, -3.23807539e-01,
              -3.88298455e-01,  5.34152471e-01, -6.45064962e-01,
               2.92092953e-01, -1.15834807e-01,  1.20258943e-01,
              -2.32704550e-01, -9.86603644e-02,  3.62706827e-01,
              -5.42650016e-01,  4.99065544e-01, -2.35553870e-01,
               6.55976624e-02,  5.41005626e-01, -6.80756752e-02,
               2.55898604e-02, -3.77840230e-01,  3.79367658e-01,
              -1.75478487e-01,  4.28158474e-02,  5.01947540e-02,
              -1.66641639e-01, -1.80162299e-02, -1.48131883e-01,
              -4.80211956e-01,  1.28889653e-01, -5.85039917e-02,
               1.97722042e-01, -1.78382062e-01,  1.76123556e-01,
              -2.80066318e-01,  3.68651973e-01, -1.58549472e-01,
               1.58607178e-01, -4.70915542e-02, -7.17698749e-02,
              -1.93604652e-02, -1.41383006e-01,  3.78527743e-01,
               6.06024379e-01, -9.69360771e-03, -7.91031146e-02,
               3.82939195e-02, -4.90137076e-02,  7.73567244e-02,
               2.51070512e-01, -3.15959680e-01,  3.79518585e-01,
               2.07648276e-01,  5.36536340e-01, -5.81876638e-01,
               4.33280793e-01, -8.84401962e-02,  1.31180870e-01,
              -1.77637390e-01,  1.87326551e-01, -1.26145397e-01,
      ...
               2.03896256e-01, -2.07276852e-01,  3.99938475e-01,
               3.81072487e-01, -6.01509056e-01,  4.04163342e-01,
              -2.54470659e-01,  2.69125823e-01,  3.27810903e-01,
              -1.98265901e-01,  1.79468204e-01, -3.32764218e-01,
              -2.12081944e-01,  3.55625899e-01,  5.61877823e-02,
               2.66856637e-01,  2.71341038e-01, -2.91072071e-01,
               4.14796262e-01, -3.77755116e-01,  5.61251626e-01,
              -2.65931885e-01,  9.03806642e-02, -5.71154398e-02,
              -4.28009549e-02, -2.18587984e-01,  2.28556500e-01,
              -1.20458900e-01,  1.93741018e-02,  7.53731510e-02,
              -2.74299903e-01,  2.83044137e-01,  1.74957018e-01,
               4.07889017e-01, -3.91677987e-01,  4.21731486e-01,
              -2.21180421e-01,  2.09137613e-01,  1.10696908e-01,
              -1.58355725e-01,  2.05783592e-01, -3.02446043e-01,
               3.16819377e-01, -1.30245528e-01, -2.46015727e-02,
              -1.43010087e-01, -9.10926214e-03, -1.87084671e-03,
               5.14908138e-02, -1.99392419e-01, -1.94496117e-01,
               5.97796497e-01, -5.67267454e-01,  4.08358419e-01,
               2.58501141e-01, -1.50884445e-02,  9.29977106e-02,
               1.67062702e-01]])

    • a_Intercept
      (chain, draw)
      float64
      1.491 1.481 1.476 ... 1.506 1.5

      array([[1.49106999, 1.48126667, 1.47639279, 1.51482428, 1.50865262,
              1.52361065, 1.48292926, 1.48089985, 1.52774428, 1.51641303,
              1.50125385, 1.51425078, 1.50399455, 1.55061734, 1.47089901,
              1.55240448, 1.57159412, 1.50486368, 1.50658531, 1.49990441,
              1.50932506, 1.53723781, 1.49345529, 1.52217964, 1.5101542 ,
              1.49267497, 1.49156222, 1.5181493 , 1.51528951, 1.50778953,
              1.51446093, 1.50199877, 1.51716078, 1.48183875, 1.52130714,
              1.48025123, 1.51693023, 1.5187197 , 1.50452056, 1.48526455,
              1.52482159, 1.51102852, 1.52950289, 1.50014217, 1.50763736,
              1.51661578, 1.52993264, 1.49116638, 1.50304885, 1.50005338,
              1.50128832, 1.51155904, 1.48586443, 1.53895175, 1.48493403,
              1.50800216, 1.48984869, 1.54463042, 1.44024817, 1.52189155,
              1.49089448, 1.50779014, 1.52601256, 1.51015058, 1.49904233,
              1.51256363, 1.47671654, 1.51203369, 1.55689484, 1.47903757,
              1.52868898, 1.52508092, 1.51234332, 1.4581175 , 1.54563815,
              1.5344382 , 1.51925963, 1.51907705, 1.51984156, 1.5609167 ,
              1.46900133, 1.53593988, 1.52054833, 1.50528245, 1.5450986 ,
              1.52347655, 1.4912987 , 1.52281441, 1.51878571, 1.46769483,
              1.58318316, 1.46443156, 1.47155848, 1.54287093, 1.54913438,
              1.49412534, 1.54690288, 1.48718757, 1.52596109, 1.49171661,
      ...
              1.50076167, 1.51780297, 1.51974274, 1.52080525, 1.53608964,
              1.52754152, 1.48424653, 1.50384339, 1.49535942, 1.52488713,
              1.51997538, 1.50393091, 1.5123751 , 1.46663312, 1.47314153,
              1.5481768 , 1.50812922, 1.54798082, 1.50850311, 1.51775067,
              1.52102535, 1.50556193, 1.48845783, 1.46705822, 1.56701742,
              1.50827543, 1.52010099, 1.50493966, 1.47618632, 1.52874861,
              1.48591924, 1.51279508, 1.49518944, 1.45728928, 1.4777127 ,
              1.53166941, 1.49077001, 1.54050944, 1.49283111, 1.5191932 ,
              1.51024845, 1.5552524 , 1.54600166, 1.47134038, 1.54726912,
              1.55164671, 1.48526564, 1.51776152, 1.50944517, 1.52056379,
              1.49427421, 1.47350097, 1.49436375, 1.49835489, 1.52026534,
              1.5234061 , 1.52846393, 1.53257015, 1.49227621, 1.51025262,
              1.48093856, 1.5024025 , 1.5068363 , 1.50261023, 1.51505796,
              1.52909602, 1.52158363, 1.54770245, 1.50065096, 1.54402857,
              1.55078994, 1.44586488, 1.46686001, 1.56368086, 1.55694375,
              1.56305391, 1.47740578, 1.56892577, 1.51783911, 1.51512358,
              1.51067504, 1.47631877, 1.51687402, 1.49315979, 1.52616865,
              1.46079157, 1.5408622 , 1.52288293, 1.51795565, 1.52488064,
              1.4821562 , 1.4936641 , 1.48250714, 1.52415042, 1.48663057,
              1.48984025, 1.50779838, 1.48306332, 1.50607781, 1.5002109 ]])

    • scaler_Intercept
      (chain, draw)
      float64
      2.639 2.517 2.512 ... 2.488 2.521

      array([[2.63910663, 2.51691166, 2.51225446, 2.50394976, 2.49235332,
              2.56861516, 2.38574936, 2.61223267, 2.46493746, 2.58239165,
              2.55697202, 2.53001329, 2.57718457, 2.5236611 , 2.50199018,
              2.49736986, 2.50914813, 2.43001644, 2.4906074 , 2.48197888,
              2.4422003 , 2.56567393, 2.45317435, 2.56923222, 2.51379831,
              2.54941103, 2.54795639, 2.4587748 , 2.45543897, 2.5120037 ,
              2.46256251, 2.53711446, 2.55840645, 2.56231227, 2.53787238,
              2.56756467, 2.51622953, 2.47767018, 2.51989225, 2.48883691,
              2.4817006 , 2.60241761, 2.51726432, 2.39061052, 2.47855292,
              2.51590895, 2.41225253, 2.61566102, 2.54498845, 2.53277367,
              2.53983564, 2.52883856, 2.60670763, 2.46493131, 2.59432252,
              2.49173373, 2.48953252, 2.5131184 , 2.45508337, 2.50749323,
              2.46401071, 2.48658291, 2.59379578, 2.52881726, 2.47950477,
              2.59406517, 2.48253403, 2.4549487 , 2.50892391, 2.4599632 ,
              2.54184659, 2.54042893, 2.53827883, 2.43320717, 2.5961008 ,
              2.57902627, 2.46523994, 2.63643408, 2.47568848, 2.50485458,
              2.51831386, 2.55490562, 2.46972471, 2.56882234, 2.53049603,
              2.61496634, 2.51093182, 2.48115869, 2.60819806, 2.4910801 ,
              2.51993507, 2.4809872 , 2.47400077, 2.54807138, 2.54834152,
              2.57873048, 2.54062806, 2.48674233, 2.56379471, 2.41539864,
      ...
              2.54053524, 2.62670257, 2.43990491, 2.56047294, 2.51053279,
              2.50139887, 2.48781993, 2.57938615, 2.49788789, 2.55662149,
              2.48907317, 2.50500347, 2.53613317, 2.39754618, 2.39938218,
              2.63577574, 2.44686925, 2.50542103, 2.47286687, 2.60810431,
              2.41884892, 2.61786983, 2.56140383, 2.57290012, 2.51591682,
              2.46159337, 2.49931475, 2.48368823, 2.55164459, 2.53763705,
              2.54598055, 2.46553472, 2.54658704, 2.56899281, 2.59250017,
              2.47813126, 2.53072736, 2.48522232, 2.49556073, 2.52319165,
              2.50802776, 2.48303132, 2.47588433, 2.47269069, 2.50820391,
              2.50587889, 2.52076347, 2.56045188, 2.51288805, 2.5014309 ,
              2.39509302, 2.47107392, 2.42420906, 2.59096186, 2.48770825,
              2.5398181 , 2.52048937, 2.52186696, 2.52165722, 2.46537344,
              2.49488954, 2.46782325, 2.52552883, 2.50495407, 2.5975604 ,
              2.43777872, 2.43884829, 2.52931248, 2.53669674, 2.5931412 ,
              2.55408799, 2.50131799, 2.56339384, 2.56337303, 2.48751249,
              2.51267927, 2.52480605, 2.58312103, 2.47648535, 2.46559084,
              2.49009109, 2.47896343, 2.56563747, 2.49321698, 2.51398487,
              2.50590937, 2.53747837, 2.57372077, 2.50462225, 2.54981397,
              2.47396595, 2.58545358, 2.44669664, 2.5782798 , 2.47926008,
              2.55740866, 2.43776369, 2.50720738, 2.48807669, 2.52110103]])

    • scaler_1|participant_id_mu
      (chain, draw)
      float64
      0.3354 -0.1478 ... -0.6267 -0.3424

      array([[ 3.35410178e-01, -1.47817388e-01, -1.29505346e-01,
              -9.95447046e-02,  1.48550878e-01, -1.45272330e-01,
               5.64389097e-02,  2.58439798e-01, -3.07121579e-01,
               2.47131656e-01, -3.74355128e-02, -4.41246391e-01,
              -3.55339841e-02, -6.61493049e-02,  7.86089491e-03,
              -3.22463118e-02,  4.26736991e-01,  3.15564964e-01,
               3.37606405e-01, -6.58557648e-02,  2.13147682e-02,
              -2.23555312e-01,  1.24784105e-01, -1.36380436e-01,
              -2.00297512e-01,  1.87740949e-01, -1.83839964e-01,
               1.51814333e-01,  1.05890321e-01, -5.39602190e-01,
              -3.76466194e-01,  3.70262959e-01,  3.74577344e-01,
              -4.83475543e-02,  2.39623042e-01,  1.87940927e-01,
              -7.76638459e-02, -8.40222570e-02,  4.23162474e-02,
               7.71533232e-02, -2.31334939e-01, -5.87415153e-02,
              -9.52340830e-03,  8.20465257e-02, -1.14552058e-01,
               5.11487117e-02, -1.18406418e-01,  1.51066429e-01,
               1.21080092e-01, -1.97537384e-03, -7.64747915e-02,
               3.56210654e-01,  1.58577843e-01, -1.97749738e-02,
              -4.23514806e-01,  5.08707966e-01, -3.93041499e-01,
              -3.81850920e-02, -2.03973566e-02,  2.81368847e-01,
      ...
               1.44441085e-01, -3.21510713e-02,  6.74630400e-02,
               1.55363320e-01,  1.05669776e-01, -2.75338606e-01,
              -6.98307271e-02,  5.07817036e-02,  1.57380665e-01,
              -1.99762656e-01,  3.70260564e-01, -3.64582950e-01,
               6.79429243e-02, -1.22806046e-01,  6.67416334e-02,
               1.55192579e-01, -7.51201938e-02,  9.64104994e-02,
              -3.95147971e-02,  2.49047437e-01, -2.70196205e-01,
               1.75321411e-02, -1.36961779e-01,  1.47138836e-01,
               1.15353525e-01, -1.02888678e-01,  9.40433211e-02,
              -1.31835053e-01,  4.61738236e-01, -5.01903674e-01,
              -2.51878586e-01,  4.62765800e-01,  2.13769223e-01,
              -1.77617406e-01,  1.92080579e-01, -2.65745592e-01,
               1.02790308e-01, -5.43633027e-02, -7.56255605e-02,
               1.83570611e-02,  1.37597633e-01, -1.57373728e-01,
               1.42195438e-01, -2.83160977e-02, -3.88231994e-02,
              -2.88198883e-01,  3.43720571e-01, -3.33420766e-01,
               8.57145697e-02,  4.62241472e-01,  7.42232390e-02,
              -7.95501228e-02,  1.11176478e-02, -6.78086935e-03,
               1.61354969e-02, -5.99075658e-01, -6.26710537e-01,
              -3.42395112e-01]])

    • rl.alpha_1|participant_id_mu
      (chain, draw)
      float64
      0.02453 0.2148 ... 0.5005 0.06875

      array([[ 2.45335207e-02,  2.14796487e-01,  1.59023778e-01,
              -1.10912834e-01,  1.31601132e-01, -8.22081262e-02,
              -9.23318902e-02,  2.37801805e-01, -2.01195126e-01,
               1.89213209e-01,  3.54341194e-01,  1.01297758e-01,
               1.28905023e-01, -1.77139008e-01,  1.70004571e-01,
              -8.97545666e-02, -2.14285317e-01, -1.97283690e-01,
              -2.57026718e-01, -3.34278552e-01,  2.33755813e-01,
               8.99006718e-02,  9.78095054e-03,  4.27738408e-02,
               5.82580380e-02,  5.60573943e-02,  2.96111786e-01,
              -1.36015221e-01, -2.83772890e-01,  1.47937333e-01,
               3.30174513e-01, -1.43387108e-02, -4.20157604e-01,
               4.87248892e-01,  9.57507793e-02, -2.35180055e-01,
               2.13393648e-01,  1.93616751e-01, -1.51811990e-01,
               5.03795716e-02,  1.14075548e-01,  3.35698096e-01,
              -2.26624828e-01,  2.58007172e-01,  1.21599887e-01,
              -4.76237923e-03,  3.09452543e-01, -2.93644097e-01,
              -1.61094179e-01,  1.47972030e-01, -2.52348129e-01,
              -1.21281158e-01, -4.94751805e-02,  1.08836690e-01,
               3.49520336e-01, -3.50835244e-01, -1.62943618e-01,
               1.68383413e-01, -2.17511853e-01,  2.90470752e-01,
      ...
              -3.96729273e-01,  5.45058489e-01, -3.76554675e-01,
              -3.89803512e-01,  5.22857532e-01, -1.58454589e-01,
               3.65187126e-01, -3.53740120e-01, -1.44002191e-01,
               1.18495608e-01, -2.38114629e-01,  3.15999115e-01,
              -4.92713461e-01,  5.46923522e-01, -4.07180066e-01,
               3.08611464e-01,  1.05023382e-01, -1.52256808e-01,
               2.56272598e-01, -3.86752326e-01,  1.97593480e-01,
              -2.67742740e-01,  3.80406775e-01, -3.26539325e-01,
              -3.21686895e-01,  3.20793758e-01, -2.41667827e-01,
               4.08886783e-02,  2.02737405e-01, -2.02496130e-01,
               5.99754904e-02, -4.93650736e-02,  4.53871752e-01,
               2.83484112e-01, -2.86883082e-01,  1.88780371e-01,
               1.31020013e-01, -1.33490429e-01, -3.50386873e-01,
              -1.06258280e-01,  9.97047710e-02, -2.00923168e-01,
               2.42069280e-01,  7.71823744e-02, -1.16731215e-01,
              -2.17011498e-03,  1.42888721e-01, -1.50911220e-01,
               6.55533714e-02, -1.02608977e-01,  2.91953460e-01,
              -2.81581167e-01,  3.51240999e-01, -3.73960994e-01,
              -1.90849644e-02, -1.15760992e-01,  5.00534695e-01,
               6.87497014e-02]])

    • theta_1|participant_id
      (chain, draw, participant_id__factor_dim)
      float64
      0.04426 -0.002328 ... 0.07588

      array([[[ 0.0442588 , -0.00232753,  0.031431  , ..., -0.01818561,
                0.00356653,  0.08318515],
              [ 0.00726152,  0.04201269,  0.0306122 , ...,  0.01435478,
               -0.00462319,  0.05228347],
              [ 0.01264699,  0.00254176,  0.00466788, ..., -0.00862629,
               -0.00057106,  0.0592016 ],
              ...,
              [ 0.04520773,  0.05298523,  0.01353253, ..., -0.05415821,
                0.07293035,  0.0855807 ],
              [ 0.00885297,  0.04116969,  0.06167496, ..., -0.04057533,
                0.01902504,  0.04201482],
              [ 0.02422518,  0.04104141,  0.02076512, ..., -0.06507524,
                0.03290414,  0.07588019]]], shape=(1, 1000, 20))

  • Indexes: (4)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

    • rl.alpha_1|participant_id__factor_dim
      PandasIndex

      PandasIndex(Index(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12',
             '13', '14', '15', '16', '17', '18', '19'],
            dtype='object', name='rl.alpha_1|participant_id__factor_dim'))

    • participant_id__factor_dim
      PandasIndex

      PandasIndex(Index(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12',
             '13', '14', '15', '16', '17', '18', '19'],
            dtype='object', name='participant_id__factor_dim'))

  • Attributes: (8)

    created_at :

    2025-07-15T19:55:45.042734+00:00

    arviz_version :

    0.22.0

    inference_library :

    numpyro

    inference_library_version :

    0.18.0

    sampling_time :

    517.168047

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• log_likelihood

  <xarray.Dataset> Size: 32MB
  Dimensions:      (chain: 1, draw: 1000, __obs__: 4000)
  Coordinates:
    * chain        (chain) int64 8B 0
    * draw         (draw) int64 8kB 0 1 2 3 4 5 6 ... 993 994 995 996 997 998 999
    * __obs__      (__obs__) int64 32kB 0 1 2 3 4 5 ... 3995 3996 3997 3998 3999
  Data variables:
      rt,response  (chain, draw, __obs__) float64 32MB -2.636 -3.071 ... -0.2937
  Attributes:
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 1000
    • __obs__: 4000
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999], shape=(1000,))

    • __obs__
      (__obs__)
      int64
      0 1 2 3 4 ... 3996 3997 3998 3999

      array([   0,    1,    2, ..., 3997, 3998, 3999], shape=(4000,))

  • Data variables: (1)
    • rt,response
      (chain, draw, __obs__)
      float64
      -2.636 -3.071 ... -0.1371 -0.2937

      array([[[-2.63575903, -3.07067751, -3.58235113, ..., -0.12729951,
               -0.1199053 , -0.36625479],
              [-3.02132081, -3.3054947 , -3.91587711, ..., -0.13660625,
               -0.16814799, -0.28142859],
              [-2.84212813, -3.19447648, -3.79152338, ..., -0.09283051,
               -0.15829979, -0.21594327],
              ...,
              [-2.76569868, -3.16024893, -3.37349719, ..., -0.08677428,
               -0.09641447, -0.32637665],
              [-2.88307955, -3.18972267, -3.43701824, ..., -0.12496733,
               -0.17307048, -0.28215016],
              [-2.77531299, -3.13847069, -3.48107947, ..., -0.07518812,
               -0.13713301, -0.29366249]]], shape=(1, 1000, 4000))

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

    • __obs__
      PandasIndex

      PandasIndex(Index([   0,    1,    2,    3,    4,    5,    6,    7,    8,    9,
             ...
             3990, 3991, 3992, 3993, 3994, 3995, 3996, 3997, 3998, 3999],
            dtype='int64', name='__obs__', length=4000))

  • Attributes: (2)

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• sample_stats

  <xarray.Dataset> Size: 57kB
  Dimensions:          (chain: 1, draw: 1000)
  Coordinates:
    * chain            (chain) int64 8B 0
    * draw             (draw) int64 8kB 0 1 2 3 4 5 6 ... 994 995 996 997 998 999
  Data variables:
      acceptance_rate  (chain, draw) float64 8kB 0.9375 0.9946 ... 0.9262 0.9207
      step_size        (chain, draw) float64 8kB 0.1562 0.1562 ... 0.1562 0.1562
      diverging        (chain, draw) bool 1kB False False False ... False True
      energy           (chain, draw) float64 8kB 4.153e+03 4.145e+03 ... 4.14e+03
      n_steps          (chain, draw) int64 8kB 31 31 31 31 31 ... 31 31 31 31 29
      tree_depth       (chain, draw) int64 8kB 5 5 5 5 5 5 5 5 ... 5 5 5 5 5 5 5 5
      lp               (chain, draw) float64 8kB 4.075e+03 4.063e+03 ... 4.057e+03
  Attributes:
      created_at:                  2025-07-15T19:55:45.056390+00:00
      arviz_version:               0.22.0
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999], shape=(1000,))

  • Data variables: (7)
    • acceptance_rate
      (chain, draw)
      float64
      0.9375 0.9946 ... 0.9262 0.9207

      array([[0.93746783, 0.99464907, 0.78335128, 0.95797434, 0.99233246,
              0.72772066, 0.8083    , 0.9098505 , 0.99226918, 0.87728014,
              0.9883639 , 0.97551034, 0.97466181, 0.92632492, 0.93055648,
              0.96786379, 0.85248314, 0.82145043, 0.83077439, 0.97432863,
              0.77482085, 0.98574552, 0.96301847, 0.98605014, 0.89544694,
              0.82491724, 0.97375321, 0.90610057, 0.80933871, 0.99569276,
              0.97246148, 0.98852382, 0.91488494, 0.92543557, 0.95542213,
              0.9459299 , 0.99035437, 0.96909479, 0.68920198, 0.9319655 ,
              0.99229207, 0.89232167, 0.98733122, 0.92868501, 0.92162471,
              0.92612254, 0.99595961, 0.96314095, 0.91445265, 0.98012361,
              0.85740421, 1.        , 0.8025675 , 0.92165831, 0.95490251,
              0.97049448, 0.95547075, 0.86717685, 0.81445293, 0.90152814,
              0.96529657, 0.8053747 , 0.84932834, 0.98925461, 0.89801942,
              0.98762315, 0.9077886 , 0.91961807, 0.9903903 , 0.97290118,
              0.96296578, 0.83357885, 0.94127187, 0.92136829, 0.98554265,
              0.86675206, 0.96389629, 0.97863609, 0.73719634, 0.91480106,
              0.75584848, 0.98375431, 0.94192359, 0.9701683 , 0.94238792,
              0.94340282, 0.86692425, 0.8734812 , 0.99761667, 0.94767621,
              0.97306596, 0.97144952, 0.99619761, 0.94286967, 0.97114099,
              0.7854196 , 0.97197201, 0.93618253, 0.94680017, 0.9555904 ,
      ...
              0.98043818, 0.8484001 , 0.92505803, 0.9964326 , 0.97734856,
              0.85332782, 0.98481506, 0.98976365, 0.97938598, 0.95581819,
              0.84783113, 0.9806777 , 0.9830957 , 0.99585434, 0.88672508,
              0.8944661 , 0.96017336, 0.9122853 , 0.84999101, 0.99580485,
              0.72727552, 0.97315627, 0.99050627, 0.91078195, 0.97123389,
              1.        , 0.72020091, 0.99532404, 0.99417339, 0.96859991,
              0.94228703, 0.94988061, 0.99282074, 0.93741108, 0.97435416,
              0.96960852, 0.98148113, 0.90450153, 0.976391  , 0.89243739,
              0.99935696, 0.94182565, 0.92183045, 0.82193333, 0.90559742,
              0.88475178, 0.98502026, 0.81614997, 0.89516061, 0.63867355,
              0.84845189, 0.86626966, 0.56598662, 0.95144555, 0.97819088,
              0.98557885, 0.92812997, 0.88290843, 0.95906395, 0.99318933,
              0.94321865, 0.96058863, 0.87465459, 0.72875762, 0.96873433,
              0.95978854, 0.67634552, 1.        , 0.89357575, 0.99802003,
              0.83017207, 0.98170094, 0.97178252, 0.69427976, 0.99476899,
              0.99094475, 0.9655324 , 0.91886361, 0.96863406, 0.99527549,
              0.93875635, 0.95747232, 0.92177572, 0.99275991, 0.93778398,
              1.        , 0.82590945, 0.71545662, 0.93001864, 0.97995158,
              0.99083529, 0.98962037, 0.97583855, 0.93055005, 0.93762783,
              0.97989826, 0.98518312, 0.83532651, 0.92622512, 0.92071524]])

    • step_size
      (chain, draw)
      float64
      0.1562 0.1562 ... 0.1562 0.1562

      array([[0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
      ...
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747,
              0.15623747, 0.15623747, 0.15623747, 0.15623747, 0.15623747]])

    • diverging
      (chain, draw)
      bool
      False False False ... False True

      array([[False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
      ...
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
               True]])

    • energy
      (chain, draw)
      float64
      4.153e+03 4.145e+03 ... 4.14e+03

      array([[4152.8043778 , 4144.82450909, 4148.38139415, 4143.00373002,
              4135.62789272, 4151.29420234, 4150.84229654, 4163.73921033,
              4148.25645363, 4141.6097248 , 4143.83514423, 4149.25498808,
              4144.94840481, 4144.92827191, 4165.85209752, 4141.08969558,
              4142.46819957, 4146.2066407 , 4151.556989  , 4141.5934863 ,
              4155.62521129, 4173.79434024, 4157.66205025, 4146.42542336,
              4156.63771842, 4163.51207784, 4162.57866429, 4150.73204961,
              4146.88713123, 4158.07157744, 4152.02721705, 4157.85610685,
              4164.21419867, 4158.04454948, 4149.57510348, 4144.97658352,
              4135.22367583, 4132.52108399, 4152.84419424, 4147.13422226,
              4147.27099577, 4148.49587291, 4141.0303175 , 4139.19560554,
              4154.95582966, 4148.96104767, 4145.82197864, 4160.83487106,
              4140.20776346, 4139.04622444, 4135.44233456, 4128.7445626 ,
              4133.28793991, 4152.3652621 , 4137.29796532, 4169.5296735 ,
              4128.30452017, 4125.32026664, 4121.75913841, 4145.00450979,
              4145.85052004, 4134.1205781 , 4147.83976539, 4153.32480314,
              4172.37621842, 4154.22906724, 4151.66596124, 4155.76953339,
              4125.67365426, 4130.05813395, 4155.60936205, 4141.29021677,
              4144.76581421, 4148.85379185, 4138.41234771, 4158.71841826,
              4125.32526164, 4106.25035815, 4123.07270723, 4122.9435605 ,
      ...
              4163.26349349, 4162.57679154, 4159.52126577, 4144.84177042,
              4158.51148266, 4161.67697789, 4161.04011587, 4158.55005567,
              4136.79194455, 4151.25838972, 4155.08074328, 4157.78185418,
              4151.20696751, 4158.15678218, 4160.30851596, 4151.91778926,
              4151.15709298, 4151.51756813, 4144.15753088, 4148.0106501 ,
              4150.37687424, 4133.6018184 , 4147.59108059, 4144.40150931,
              4132.95867153, 4127.58480853, 4116.96534015, 4113.48178631,
              4119.92024489, 4119.09396547, 4122.86163494, 4129.59772393,
              4151.40018017, 4150.0226073 , 4153.32484054, 4153.3877969 ,
              4152.24479255, 4146.73868101, 4154.9585202 , 4137.78987174,
              4137.81245257, 4137.39716872, 4141.30812254, 4161.30752083,
              4158.28823149, 4140.88702537, 4123.65234526, 4111.75117677,
              4128.16158674, 4147.3880296 , 4155.28993367, 4154.15318667,
              4153.58422458, 4159.13655021, 4164.15509408, 4162.23178063,
              4154.71696997, 4145.47749341, 4165.57231749, 4137.49563836,
              4143.68576578, 4136.85823865, 4140.85794623, 4136.11096117,
              4131.60103294, 4139.92126096, 4161.41702492, 4150.06268103,
              4155.57065986, 4157.26121126, 4161.65883374, 4148.00651276,
              4169.69908783, 4169.70707514, 4158.70332378, 4130.59303241,
              4136.65251942, 4136.40023908, 4144.01760984, 4140.38566453]])

    • n_steps
      (chain, draw)
      int64
      31 31 31 31 31 ... 31 31 31 31 29

      array([[31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
      ...
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31,
              31, 31, 31, 31, 31, 31, 31, 29]])

    • tree_depth
      (chain, draw)
      int64
      5 5 5 5 5 5 5 5 ... 5 5 5 5 5 5 5 5

      array([[5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
      ...
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
              5, 5, 5, 5, 5, 5, 5, 5, 5, 5]])

    • lp
      (chain, draw)
      float64
      4.075e+03 4.063e+03 ... 4.057e+03

      array([[4074.62573622, 4062.73964321, 4069.04981151, 4052.74611852,
              4061.49849071, 4063.76922335, 4076.52523457, 4069.63948177,
              4060.75646519, 4072.3588423 , 4060.49813797, 4065.79246532,
              4063.16854087, 4071.59674997, 4064.82980337, 4056.62942264,
              4064.9806756 , 4054.78809698, 4070.44897698, 4060.33148408,
              4077.51648719, 4085.46132927, 4064.66252907, 4051.24042259,
              4074.11699965, 4064.98687284, 4075.20997611, 4051.33078833,
              4081.38573257, 4065.85607551, 4064.2621233 , 4075.71027112,
              4080.96269555, 4071.13762255, 4059.69651828, 4065.75218129,
              4057.47506638, 4057.20257234, 4059.94645279, 4074.59612031,
              4064.79020691, 4065.8729987 , 4057.37842782, 4072.17122322,
              4079.68257039, 4063.20596645, 4078.19261204, 4070.52254139,
              4056.86512261, 4047.03277394, 4063.71997667, 4051.34137474,
              4059.37530636, 4065.77500407, 4068.534483  , 4062.23478438,
              4055.31937598, 4047.01061467, 4052.9682814 , 4056.15681765,
              4048.17533265, 4063.72531741, 4075.72255712, 4082.28857625,
              4083.71225707, 4065.34432185, 4074.03881977, 4061.03760539,
              4042.67357799, 4066.41366375, 4076.70043186, 4055.57243948,
              4063.74244156, 4071.33358969, 4068.59915468, 4065.31502753,
              4039.41083753, 4037.32088529, 4053.35503398, 4048.7368359 ,
      ...
              4073.40081066, 4083.05433416, 4066.59497343, 4058.40496848,
              4074.28216462, 4068.45256453, 4082.63696528, 4067.14798171,
              4071.54429212, 4070.23170823, 4069.94825493, 4080.43863771,
              4066.74505006, 4073.87089853, 4069.62870336, 4072.01673198,
              4064.61407622, 4070.71683214, 4063.0375435 , 4070.4545854 ,
              4061.09782543, 4063.00378205, 4056.9676769 , 4050.79902432,
              4049.83929646, 4046.86067898, 4039.39888465, 4036.85780105,
              4046.13053548, 4050.08868334, 4047.49852955, 4050.17693638,
              4062.76276266, 4071.67591108, 4064.34227061, 4070.74614367,
              4067.84755938, 4070.03302864, 4073.71720689, 4058.71332424,
              4051.77491159, 4058.48672421, 4063.80089438, 4077.52457397,
              4065.58202859, 4044.53210322, 4049.76922554, 4036.58435827,
              4072.27188282, 4051.18929634, 4075.64083177, 4063.98071104,
              4074.98253961, 4087.76773911, 4078.88079791, 4078.09593957,
              4072.00746249, 4080.32637059, 4066.14755021, 4065.84437144,
              4063.16836039, 4065.2837993 , 4056.16981674, 4062.35564929,
              4059.58475247, 4067.56560861, 4056.73727462, 4082.29372279,
              4067.00874284, 4081.50930878, 4070.57738867, 4076.47623521,
              4068.19043575, 4079.6334369 , 4063.85776269, 4056.7369128 ,
              4051.96607075, 4057.50926366, 4062.42680635, 4057.27964191]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (4)

    created_at :

    2025-07-15T19:55:45.056390+00:00

    arviz_version :

    0.22.0

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• observed_data

  <xarray.Dataset> Size: 96kB
  Dimensions:                  (__obs__: 4000, rt,response_extra_dim_0: 2)
  Coordinates:
    * __obs__                  (__obs__) int64 32kB 0 1 2 3 ... 3997 3998 3999
    * rt,response_extra_dim_0  (rt,response_extra_dim_0) int64 16B 0 1
  Data variables:
      rt,response              (__obs__, rt,response_extra_dim_0) float64 64kB ...
  Attributes:
      created_at:                  2025-07-15T19:55:45.057017+00:00
      arviz_version:               0.22.0
      inference_library:           numpyro
      inference_library_version:   0.18.0
      sampling_time:               517.168047
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • __obs__: 4000
    • rt,response_extra_dim_0: 2
  • Coordinates: (2)
    • __obs__
      (__obs__)
      int64
      0 1 2 3 4 ... 3996 3997 3998 3999

      array([   0,    1,    2, ..., 3997, 3998, 3999], shape=(4000,))

    • rt,response_extra_dim_0
      (rt,response_extra_dim_0)
      int64
      0 1

      array([0, 1])

  • Data variables: (1)
    • rt,response
      (__obs__, rt,response_extra_dim_0)
      float64
      0.9356 0.0 1.114 ... 1.0 1.197 1.0

      array([[0.93560183, 0.        ],
             [1.11437929, 0.        ],
             [0.56431085, 0.        ],
             ...,
             [1.32044899, 1.        ],
             [1.39262187, 1.        ],
             [1.19681287, 1.        ]], shape=(4000, 2))

  • Indexes: (2)
    • __obs__
      PandasIndex

      PandasIndex(Index([   0,    1,    2,    3,    4,    5,    6,    7,    8,    9,
             ...
             3990, 3991, 3992, 3993, 3994, 3995, 3996, 3997, 3998, 3999],
            dtype='int64', name='__obs__', length=4000))

    • rt,response_extra_dim_0
      PandasIndex

      PandasIndex(Index([0, 1], dtype='int64', name='rt,response_extra_dim_0'))

  • Attributes: (8)

    created_at :

    2025-07-15T19:55:45.057017+00:00

    arviz_version :

    0.22.0

    inference_library :

    numpyro

    inference_library_version :

    0.18.0

    sampling_time :

    517.168047

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  

Assess the model fits¶

We examine the quality of fits by comparing the recovered parameters with the ground-truth data generating parameters of the simulated dataset. We examine the quality of fits both at group-level as well as subject-level.

Examining group-level posteriors¶

In [12]:

# Define parameter names for analysis

list_group_mean_params = [
    "rl.alpha_Intercept",
    "rl.alpha_neg_Intercept",
    "scaler_Intercept",
    "a_Intercept",
    "z_Intercept",
    "t_Intercept",
    "theta_Intercept",
]

list_group_sd_params = [
    "rl.alpha_1|participant_id_sigma",
    "rl.alpha_neg_1|participant_id_sigma",
    "scaler_1|participant_id_sigma",
    "a_1|participant_id_sigma", 
    "z_1|participant_id_sigma",
    "t_1|participant_id_sigma",
    "theta_1|participant_id_sigma",
]

# Define parameter names for analysis list_group_mean_params = [ "rl.alpha_Intercept", "rl.alpha_neg_Intercept", "scaler_Intercept", "a_Intercept", "z_Intercept", "t_Intercept", "theta_Intercept", ] list_group_sd_params = [ "rl.alpha_1|participant_id_sigma", "rl.alpha_neg_1|participant_id_sigma", "scaler_1|participant_id_sigma", "a_1|participant_id_sigma", "z_1|participant_id_sigma", "t_1|participant_id_sigma", "theta_1|participant_id_sigma", ]

In [13]:

# Create mapping from HSSM model parameter names to ground truth values.
def create_ground_truth_mapping(savefile):

    return {
    "rl.alpha_Intercept": savefile['params_true_group']['rl_alpha_mean'],
    "scaler_Intercept": savefile['params_true_group']['scaler_mean'],
    "a_Intercept": savefile['params_true_group']['a_mean'],
    "z_Intercept": savefile['params_true_group']['z_mean'],
    "t_Intercept": savefile['params_true_group']['t_mean'],
    "theta_Intercept": savefile['params_true_group']['theta_mean'],
}

ground_truth_params = create_ground_truth_mapping(savefile)
print("Ground truth group means:\n")
for param, value in ground_truth_params.items():
    print(f"{param}: {value:.3f}")

# Create mapping from HSSM model parameter names to ground truth values. def create_ground_truth_mapping(savefile): return { "rl.alpha_Intercept": savefile['params_true_group']['rl_alpha_mean'], "scaler_Intercept": savefile['params_true_group']['scaler_mean'], "a_Intercept": savefile['params_true_group']['a_mean'], "z_Intercept": savefile['params_true_group']['z_mean'], "t_Intercept": savefile['params_true_group']['t_mean'], "theta_Intercept": savefile['params_true_group']['theta_mean'], } ground_truth_params = create_ground_truth_mapping(savefile) print("Ground truth group means:\n") for param, value in ground_truth_params.items(): print(f"{param}: {value:.3f}")

Ground truth group means:

rl.alpha_Intercept: 0.660
scaler_Intercept: 2.785
a_Intercept: 1.481
z_Intercept: 0.267
t_Intercept: 0.443
theta_Intercept: 0.312

In [14]:

# Plot posterior distributions and MCMC traces for group-level parameters
# Vertical lines show ground truth values for parameter recovery assessment
az.plot_trace(idata_mcmc, var_names=list_group_mean_params,
              lines=[(key_, {}, ground_truth_params[key_]) for key_ in ground_truth_params])
plt.tight_layout()

# Plot posterior distributions and MCMC traces for group-level parameters # Vertical lines show ground truth values for parameter recovery assessment az.plot_trace(idata_mcmc, var_names=list_group_mean_params, lines=[(key_, {}, ground_truth_params[key_]) for key_ in ground_truth_params]) plt.tight_layout()

In [ ]: