Colab Instructions¶

If you would like to run this tutorial on Google colab, please click this link.

Once you are in the colab, follow the installation instructions below and then restart your runtime.

Just uncomment the code in the next code cell and run it!

NOTE:

You may want to switch your runtime to have a GPU or TPU. To do so, go to Runtime > Change runtime type and select the desired hardware accelerator.

Note that if you switch your runtime you have to follow the installation instructions again.

In [1]:

# If running this on Colab, please uncomment the next line
# !pip install hssm

# If running this on Colab, please uncomment the next line # !pip install hssm

Using lapse probabilities and distributions in HSSM¶

Since v0.1.2, HSSM has added the ability to model outliers in the distribution with lapse probabilities and distributions. Let's see how it works.

In [2]:

import bambi as bmb
import matplotlib.pyplot as plt
import hssm
import warnings

warnings.filterwarnings("ignore")

import bambi as bmb import matplotlib.pyplot as plt import hssm import warnings warnings.filterwarnings("ignore")

Lapse probabilities and distributions are enabled by default.¶

From v0.1.2 on, lapse probabilities and distributions are enabled by default. If left unspecified, the probability for outliers will be a fixed value of 0.05 and the distribution will be specified as Uniform(0, 10).

In [3]:

# Simulate some data
ddm_data = hssm.simulate_data(
    model="ddm", theta=dict(v=0.5, a=1.5, z=0.5, t=0.1), size=1000
)
ddm_data.head()

# Simulate some data ddm_data = hssm.simulate_data( model="ddm", theta=dict(v=0.5, a=1.5, z=0.5, t=0.1), size=1000 ) ddm_data.head()

Out[3]:

	rt	response
0	0.738123	-1.0
1	3.669316	1.0
2	1.302614	1.0
3	1.763521	-1.0
4	2.213896	1.0

In [4]:

# Build the simplest model specifying only data
# Note the model output

ddm_model_default = hssm.HSSM(data=ddm_data)
ddm_model_default

# Build the simplest model specifying only data # Note the model output ddm_model_default = hssm.HSSM(data=ddm_data) ddm_model_default

Model initialized successfully.

Out[4]:

Hierarchical Sequential Sampling Model
Model: ddm

Response variable: rt,response
Likelihood: analytical
Observations: 1000

Parameters:

v:
    Prior: Normal(mu: 0.0, sigma: 2.0)
    Explicit bounds: (-inf, inf)

a:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

z:
    Prior: Uniform(lower: 0.0, upper: 1.0)
    Explicit bounds: (0.0, 1.0)

t:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)


Lapse probability: 0.05
Lapse distribution: Uniform(lower: 0.0, upper: 20.0)

Note

Note in the above output that lapse probability and lapse distributions are at default values.

Specifying lapse probability and distribution¶

It is easy to change the lapse probability and distribution. HSSM has added two arguments, p_outlier and lapse to allow the lapse probability and distribution to be specified.

The optional p_outlier accepts a float, a dict, or a bmb.Prior object. When p_outlier is specified as a single float value, it will be considered "fixed". You can also specify a prior distribution for p_outlier through a dict or a bmb.Prior object, the same way as you would when specifying priors for any other parameter. That way, the lapse probability will be considered another parameter and will be estimated during MCMC sampling.

Likewise, the lapse argument accepts a dict or a bmb.Prior object to specify a fixed lapse distribution. This distribution will be considered as the one that outliers are generated from.

In [5]:

ddm_model_lapse = hssm.HSSM(
    data=ddm_data,
    p_outlier={"name": "Uniform", "lower": 0.0001, "upper": 0.5},
    lapse=bmb.Prior("Uniform", lower=0.0, upper=20.0),
)

ddm_model_lapse

ddm_model_lapse = hssm.HSSM( data=ddm_data, p_outlier={"name": "Uniform", "lower": 0.0001, "upper": 0.5}, lapse=bmb.Prior("Uniform", lower=0.0, upper=20.0), ) ddm_model_lapse

Model initialized successfully.

Out[5]:

Hierarchical Sequential Sampling Model
Model: ddm

Response variable: rt,response
Likelihood: analytical
Observations: 1000

Parameters:

v:
    Prior: Normal(mu: 0.0, sigma: 2.0)
    Explicit bounds: (-inf, inf)

a:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

z:
    Prior: Uniform(lower: 0.0, upper: 1.0)
    Explicit bounds: (0.0, 1.0)

t:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)


Lapse probability: Uniform(lower: 0.0001, upper: 0.5)
Lapse distribution: Uniform(lower: 0.0, upper: 20.0)

In [6]:

lapse_trace = ddm_model_lapse.sample()
lapse_trace

lapse_trace = ddm_model_lapse.sample() lapse_trace

Using default initvals. 

Initializing NUTS using adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [z, p_outlier, t, a, v]

Output()

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 23 seconds.
100%|██████████| 4000/4000 [00:01<00:00, 2654.19it/s]

Out[6]:

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 168kB
  Dimensions:    (chain: 4, draw: 1000)
  Coordinates:
    * chain      (chain) int64 32B 0 1 2 3
    * draw       (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
  Data variables:
      p_outlier  (chain, draw) float64 32kB 0.01574 0.0164 ... 0.01012 0.01252
      t          (chain, draw) float64 32kB 0.09827 0.1512 ... 0.1614 0.1558
      z          (chain, draw) float64 32kB 0.4651 0.4722 0.4727 ... 0.4891 0.4993
      a          (chain, draw) float64 32kB 1.571 1.454 1.523 ... 1.435 1.449
      v          (chain, draw) float64 32kB 0.6329 0.5217 0.5652 ... 0.5189 0.5327
  Attributes:
      created_at:                  2025-07-13T13:58:06.293676+00:00
      arviz_version:               0.21.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               23.291183948516846
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

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

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

  • Data variables: (5)
    • p_outlier
      (chain, draw)
      float64
      0.01574 0.0164 ... 0.01012 0.01252

      array([[0.01573516, 0.01639739, 0.01286207, ..., 0.02002668, 0.02002668,
              0.01389091],
             [0.00791601, 0.00665491, 0.00911218, ..., 0.01738749, 0.01609305,
              0.01686079],
             [0.00396808, 0.02197937, 0.01208184, ..., 0.01244974, 0.00391847,
              0.00975214],
             [0.00787461, 0.01007891, 0.00842142, ..., 0.01029489, 0.0101151 ,
              0.01252234]])

    • t
      (chain, draw)
      float64
      0.09827 0.1512 ... 0.1614 0.1558

      array([[0.09826979, 0.15122511, 0.11068024, ..., 0.1674854 , 0.1674854 ,
              0.14486555],
             [0.12582787, 0.10540166, 0.15415957, ..., 0.13207355, 0.1457792 ,
              0.13424558],
             [0.13560738, 0.13744356, 0.14641952, ..., 0.1430671 , 0.13981405,
              0.1279039 ],
             [0.14223858, 0.14876673, 0.13482299, ..., 0.12747666, 0.16141789,
              0.15583025]])

    • z
      (chain, draw)
      float64
      0.4651 0.4722 ... 0.4891 0.4993

      array([[0.46510178, 0.47216061, 0.47273151, ..., 0.49337591, 0.49337591,
              0.494426  ],
             [0.48396161, 0.48690385, 0.48280956, ..., 0.48557662, 0.48376786,
              0.49082977],
             [0.48615678, 0.48593228, 0.4663094 , ..., 0.47488228, 0.49274161,
              0.47180312],
             [0.49290801, 0.49241407, 0.48143615, ..., 0.47806048, 0.48907334,
              0.49929144]])

    • a
      (chain, draw)
      float64
      1.571 1.454 1.523 ... 1.435 1.449

      array([[1.57093198, 1.4539095 , 1.52336542, ..., 1.49197004, 1.49197004,
              1.47748158],
             [1.54604963, 1.48810738, 1.5335453 , ..., 1.46967294, 1.49307487,
              1.49183604],
             [1.49874716, 1.45771579, 1.45924196, ..., 1.47388511, 1.47778324,
              1.49884755],
             [1.44182808, 1.47563146, 1.48484313, ..., 1.49251346, 1.43490113,
              1.44885929]])

    • v
      (chain, draw)
      float64
      0.6329 0.5217 ... 0.5189 0.5327

      array([[0.63289966, 0.52167145, 0.56521554, ..., 0.55257481, 0.55257481,
              0.57272428],
             [0.56325267, 0.51339017, 0.59076093, ..., 0.58793742, 0.54979776,
              0.55571528],
             [0.55809953, 0.51640256, 0.63625737, ..., 0.60549531, 0.533846  ,
              0.58686603],
             [0.47605032, 0.56516109, 0.5894369 , ..., 0.54204857, 0.51888448,
              0.53267793]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], 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: (8)

    created_at :

    2025-07-13T13:58:06.293676+00:00

    arviz_version :

    0.21.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    23.291183948516846

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• log_likelihood

  <xarray.Dataset> Size: 32MB
  Dimensions:      (chain: 4, draw: 1000, __obs__: 1000)
  Coordinates:
    * chain        (chain) int64 32B 0 1 2 3
    * draw         (draw) int64 8kB 0 1 2 3 4 5 6 ... 993 994 995 996 997 998 999
    * __obs__      (__obs__) int64 8kB 0 1 2 3 4 5 6 ... 994 995 996 997 998 999
  Data variables:
      rt,response  (chain, draw, __obs__) float64 32MB -2.597 -2.595 ... -1.806
  Attributes:
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
    • __obs__: 1000
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

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

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

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

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

  • Data variables: (1)
    • rt,response
      (chain, draw, __obs__)
      float64
      -2.597 -2.595 ... -5.189 -1.806

      array([[[-2.59672243, -2.59504955, -1.04379997, ..., -1.07321863,
               -5.34933012, -1.67808923],
              [-2.21362384, -2.72979653, -1.0977449 , ..., -1.0407602 ,
               -5.08167333, -1.78140572],
              [-2.42794551, -2.64440143, -1.08605102, ..., -1.05371559,
               -5.17966601, -1.73355874],
              ...,
              [-2.49836562, -2.68771856, -1.0590188 , ..., -0.98583761,
               -5.11607742, -1.75287038],
              [-2.49836562, -2.68771856, -1.0590188 , ..., -0.98583761,
               -5.11607742, -1.75287038],
              [-2.49521686, -2.74000948, -1.04908843, ..., -0.90419002,
               -5.27462064, -1.77251079]],
      
             [[-2.55157517, -2.60988662, -1.07344886, ..., -1.05214084,
               -5.18700279, -1.71611685],
              [-2.35140006, -2.71068094, -1.12109062, ..., -0.98675205,
               -5.10991361, -1.79538518],
              [-2.57686141, -2.62219348, -1.03289874, ..., -1.03989556,
               -5.28255135, -1.69684666],
      ...
              [-2.41269943, -2.73588985, -1.02562898, ..., -0.94588225,
               -5.38984143, -1.74350152],
              [-2.40941508, -2.72055775, -1.07721419, ..., -0.94549373,
               -5.21272303, -1.77770988],
              [-2.40984769, -2.68396441, -1.05152718, ..., -1.00621346,
               -5.31206374, -1.73147581]],
      
             [[-2.23436319, -2.77207234, -1.12799791, ..., -0.96376754,
               -5.03586499, -1.83172806],
              [-2.46340208, -2.7328728 , -1.04995206, ..., -0.91957346,
               -5.27788406, -1.76789209],
              [-2.44864139, -2.72060171, -1.03963273, ..., -0.94055433,
               -5.36533322, -1.74856625],
              ...,
              [-2.35901485, -2.68577487, -1.0893703 , ..., -1.02154013,
               -5.1562711 , -1.75900184],
              [-2.27338163, -2.77989819, -1.08029019, ..., -0.93652055,
               -5.16759663, -1.80548308],
              [-2.39483224, -2.7761184 , -1.07263547, ..., -0.89383182,
               -5.18867828, -1.80621528]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], 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,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='__obs__', length=1000))

  • Attributes: (2)

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• sample_stats

  <xarray.Dataset> Size: 496kB
  Dimensions:                (chain: 4, draw: 1000)
  Coordinates:
    * chain                  (chain) int64 32B 0 1 2 3
    * draw                   (draw) int64 8kB 0 1 2 3 4 5 ... 995 996 997 998 999
  Data variables: (12/17)
      n_steps                (chain, draw) float64 32kB 7.0 7.0 7.0 ... 7.0 3.0
      tree_depth             (chain, draw) int64 32kB 3 3 3 4 3 3 ... 4 3 4 3 3 2
      largest_eigval         (chain, draw) float64 32kB nan nan nan ... nan nan
      perf_counter_diff      (chain, draw) float64 32kB 0.007566 ... 0.003679
      acceptance_rate        (chain, draw) float64 32kB 0.6021 0.8763 ... 0.9702
      index_in_trajectory    (chain, draw) int64 32kB 7 -6 6 6 3 -3 ... -3 3 1 2 2
      ...                     ...
      energy_error           (chain, draw) float64 32kB 0.6817 0.1329 ... -0.02246
      diverging              (chain, draw) bool 4kB False False ... False False
      process_time_diff      (chain, draw) float64 32kB 0.007476 ... 0.003676
      lp                     (chain, draw) float64 32kB -1.948e+03 ... -1.944e+03
      step_size              (chain, draw) float64 32kB 0.6981 0.6981 ... 0.5802
      perf_counter_start     (chain, draw) float64 32kB 1.433e+06 ... 1.433e+06
  Attributes:
      created_at:                  2025-07-13T13:58:06.312112+00:00
      arviz_version:               0.21.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               23.291183948516846
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

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

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

  • Data variables: (17)
    • n_steps
      (chain, draw)
      float64
      7.0 7.0 7.0 11.0 ... 7.0 7.0 3.0

      array([[ 7.,  7.,  7., ...,  7.,  1.,  3.],
             [15.,  7.,  7., ...,  7.,  7., 15.],
             [ 7.,  7.,  7., ...,  7.,  7.,  7.],
             [15.,  3.,  7., ...,  7.,  7.,  3.]])

    • tree_depth
      (chain, draw)
      int64
      3 3 3 4 3 3 3 3 ... 2 3 4 3 4 3 3 2

      array([[3, 3, 3, ..., 3, 1, 2],
             [4, 3, 3, ..., 3, 3, 4],
             [3, 3, 3, ..., 3, 3, 3],
             [4, 2, 3, ..., 3, 3, 2]])

    • largest_eigval
      (chain, draw)
      float64
      nan nan nan nan ... nan nan nan nan

      array([[nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan]])

    • perf_counter_diff
      (chain, draw)
      float64
      0.007566 0.007286 ... 0.003679

      array([[0.00756583, 0.00728633, 0.00766017, ..., 0.0074745 , 0.001815  ,
              0.00384338],
             [0.01535079, 0.00798162, 0.00747388, ..., 0.00740762, 0.00739558,
              0.01440625],
             [0.00862633, 0.00737296, 0.0079635 , ..., 0.00738854, 0.00792658,
              0.00743921],
             [0.01456529, 0.00363937, 0.00745133, ..., 0.00702842, 0.00735171,
              0.00367904]])

    • acceptance_rate
      (chain, draw)
      float64
      0.6021 0.8763 1.0 ... 0.9055 0.9702

      array([[0.60209745, 0.87632624, 1.        , ..., 0.69849304, 0.25433661,
              0.93641742],
             [1.        , 0.89074653, 0.74484331, ..., 1.        , 0.93183466,
              0.98488285],
             [0.8611644 , 0.84022089, 0.96305824, ..., 0.93617908, 0.97572825,
              0.97488619],
             [0.88079655, 0.96651238, 0.95202732, ..., 0.86459918, 0.90548607,
              0.97016777]])

    • index_in_trajectory
      (chain, draw)
      int64
      7 -6 6 6 3 -3 4 ... 4 -5 -3 3 1 2 2

      array([[ 7, -6,  6, ...,  1,  0,  1],
             [ 9, -3,  3, ...,  3, -7,  1],
             [ 2,  5, -3, ...,  5, -6,  2],
             [ 6,  2, -6, ...,  1,  2,  2]])

    • max_energy_error
      (chain, draw)
      float64
      0.6817 0.4072 ... 0.3109 0.09376

      array([[ 0.68167611,  0.40720218, -0.80618835, ...,  1.07681464,
               1.36909664, -0.36010792],
             [-1.16399699,  0.37540623,  0.92631092, ..., -0.73872395,
              -0.43742755, -0.2825444 ],
             [ 0.39846135,  0.3122735 , -0.2555816 , ..., -0.38710646,
               0.12204308,  0.17254765],
             [ 0.4086284 , -0.3157872 ,  0.14487813, ...,  0.46604523,
               0.31085492,  0.09375774]])

    • smallest_eigval
      (chain, draw)
      float64
      nan nan nan nan ... nan nan nan nan

      array([[nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan],
             [nan, nan, nan, ..., nan, nan, nan]])

    • step_size_bar
      (chain, draw)
      float64
      0.5501 0.5501 ... 0.495 0.495

      array([[0.55005344, 0.55005344, 0.55005344, ..., 0.55005344, 0.55005344,
              0.55005344],
             [0.52428984, 0.52428984, 0.52428984, ..., 0.52428984, 0.52428984,
              0.52428984],
             [0.58275057, 0.58275057, 0.58275057, ..., 0.58275057, 0.58275057,
              0.58275057],
             [0.49501154, 0.49501154, 0.49501154, ..., 0.49501154, 0.49501154,
              0.49501154]])

    • reached_max_treedepth
      (chain, draw)
      bool
      False False False ... False False

      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]])

    • energy
      (chain, draw)
      float64
      1.949e+03 1.95e+03 ... 1.945e+03

      array([[1948.77918451, 1949.83556068, 1945.97748186, ..., 1950.95382267,
              1948.43019797, 1945.83547038],
             [1948.72466236, 1947.2752532 , 1949.39537237, ..., 1945.32081988,
              1944.31393947, 1942.93968212],
             [1944.20087756, 1945.93640849, 1946.73742328, ..., 1944.81339559,
              1944.93716913, 1945.65713306],
             [1949.0154015 , 1947.36005806, 1943.62011978, ..., 1945.25484297,
              1947.41356362, 1944.89169829]])

    • energy_error
      (chain, draw)
      float64
      0.6817 0.1329 ... -0.01551 -0.02246

      array([[ 0.68167611,  0.13289886, -0.66823419, ..., -0.02805843,
               0.        , -0.36010792],
             [-0.73640024, -0.11479816,  0.92631092, ..., -0.4911664 ,
              -0.19469002, -0.23197585],
             [-0.08454396,  0.14576169,  0.19111761, ..., -0.20783373,
              -0.01444468,  0.01746653],
             [ 0.39796849, -0.3157872 ,  0.11847156, ..., -0.14311705,
              -0.01550896, -0.02245877]])

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

      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]])

    • process_time_diff
      (chain, draw)
      float64
      0.007476 0.007268 ... 0.003676

      array([[0.007476, 0.007268, 0.007578, ..., 0.007452, 0.001816, 0.003759],
             [0.01515 , 0.007745, 0.007457, ..., 0.00738 , 0.007386, 0.014399],
             [0.007859, 0.007358, 0.007779, ..., 0.007378, 0.007702, 0.007425],
             [0.014526, 0.003639, 0.007437, ..., 0.007028, 0.007351, 0.003676]])

    • lp
      (chain, draw)
      float64
      -1.948e+03 ... -1.944e+03

      array([[-1948.08810115, -1945.44646547, -1943.44131494, ...,
              -1945.08971985, -1945.08971985, -1942.92246568],
             [-1945.58572008, -1944.97057013, -1948.23147276, ...,
              -1943.04972894, -1942.71911499, -1942.14170183],
             [-1943.27978868, -1944.04882257, -1945.79080332, ...,
              -1943.05427343, -1943.28481739, -1942.5482654 ],
             [-1946.55556912, -1942.68962323, -1942.83564207, ...,
              -1942.5225083 , -1944.25025817, -1943.5019777 ]])

    • step_size
      (chain, draw)
      float64
      0.6981 0.6981 ... 0.5802 0.5802

      array([[0.69807644, 0.69807644, 0.69807644, ..., 0.69807644, 0.69807644,
              0.69807644],
             [0.40125009, 0.40125009, 0.40125009, ..., 0.40125009, 0.40125009,
              0.40125009],
             [0.5865143 , 0.5865143 , 0.5865143 , ..., 0.5865143 , 0.5865143 ,
              0.5865143 ],
             [0.58016557, 0.58016557, 0.58016557, ..., 0.58016557, 0.58016557,
              0.58016557]])

    • perf_counter_start
      (chain, draw)
      float64
      1.433e+06 1.433e+06 ... 1.433e+06

      array([[1432733.03893033, 1432733.04657371, 1432733.05392017, ...,
              1432742.15587687, 1432742.16341667, 1432742.16527554],
             [1432734.57324279, 1432734.58869525, 1432734.59675871, ...,
              1432743.1512345 , 1432743.15871388, 1432743.16619562],
             [1432733.83977979, 1432733.84849471, 1432733.85593288, ...,
              1432742.37185387, 1432742.37931929, 1432742.38731638],
             [1432735.06743496, 1432735.08209296, 1432735.08578596, ...,
              1432743.90455867, 1432743.91163279, 1432743.91903888]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], 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: (8)

    created_at :

    2025-07-13T13:58:06.312112+00:00

    arviz_version :

    0.21.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    23.291183948516846

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• observed_data

  <xarray.Dataset> Size: 24kB
  Dimensions:                  (__obs__: 1000, rt,response_extra_dim_0: 2)
  Coordinates:
    * __obs__                  (__obs__) int64 8kB 0 1 2 3 4 ... 996 997 998 999
    * 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 16kB ...
  Attributes:
      created_at:                  2025-07-13T13:58:06.314769+00:00
      arviz_version:               0.21.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • __obs__: 1000
    • rt,response_extra_dim_0: 2
  • Coordinates: (2)
    • __obs__
      (__obs__)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

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

    • 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
      0.7381 -1.0 3.669 ... 2.331 1.0

      array([[ 0.73812306, -1.        ],
             [ 3.66931605,  1.        ],
             [ 1.30261397,  1.        ],
             ...,
             [ 0.68580204,  1.        ],
             [ 5.00976229, -1.        ],
             [ 2.3309052 ,  1.        ]])

  • Indexes: (2)
    • __obs__
      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='__obs__', length=1000))

    • rt,response_extra_dim_0
      PandasIndex

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

  • Attributes: (6)

    created_at :

    2025-07-13T13:58:06.314769+00:00

    arviz_version :

    0.21.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  

In [7]:

ddm_model_lapse.summary(lapse_trace)

ddm_model_lapse.summary(lapse_trace)

Out[7]:

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
t	0.129	0.020	0.093	0.168	0.000	0.000	2236.0	2008.0	1.0
v	0.564	0.035	0.497	0.628	0.001	0.001	2337.0	2251.0	1.0
p_outlier	0.013	0.008	0.000	0.029	0.000	0.000	2150.0	1410.0	1.0
z	0.481	0.013	0.458	0.505	0.000	0.000	2335.0	2380.0	1.0
a	1.488	0.027	1.438	1.540	0.001	0.000	2263.0	2411.0	1.0

In [8]:

ddm_model_lapse.plot_trace(lapse_trace)
plt.tight_layout()

ddm_model_lapse.plot_trace(lapse_trace) plt.tight_layout()

Disable lapse probabilities and distributions¶

When p_outliers is set to None or 0, lapse probability and distributions will be ignored. They will not be included in the model output.

Note

If `p_outlier` is set to `None` or 0 but `lapse` is not set to None, you will receive a warning. Please remember to also set `lapse` to `None`.

In [9]:

ddm_model_no_lapse = hssm.HSSM(data=ddm_data, p_outlier=None, lapse=None)
ddm_model_no_lapse

ddm_model_no_lapse = hssm.HSSM(data=ddm_data, p_outlier=None, lapse=None) ddm_model_no_lapse

Model initialized successfully.

Out[9]:

Hierarchical Sequential Sampling Model
Model: ddm

Response variable: rt,response
Likelihood: analytical
Observations: 1000

Parameters:

v:
    Prior: Normal(mu: 0.0, sigma: 2.0)
    Explicit bounds: (-inf, inf)

a:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

z:
    Prior: Uniform(lower: 0.0, upper: 1.0)
    Explicit bounds: (0.0, 1.0)

t:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

In [10]:

ddm_model_no_lapse_warn = hssm.HSSM(data=ddm_data, p_outlier=0)
ddm_model_no_lapse_warn

ddm_model_no_lapse_warn = hssm.HSSM(data=ddm_data, p_outlier=0) ddm_model_no_lapse_warn

You have specified the `lapse` argument to include a lapse distribution, but `p_outlier` is set to either 0 or None. Your lapse distribution will be ignored.
Model initialized successfully.

Out[10]:

Hierarchical Sequential Sampling Model
Model: ddm

Response variable: rt,response
Likelihood: analytical
Observations: 1000

Parameters:

v:
    Prior: Normal(mu: 0.0, sigma: 2.0)
    Explicit bounds: (-inf, inf)

a:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)

z:
    Prior: Uniform(lower: 0.0, upper: 1.0)
    Explicit bounds: (0.0, 1.0)

t:
    Prior: HalfNormal(sigma: 2.0)
    Explicit bounds: (0.0, inf)