Wald response times, Racing diffusion model

Diffusion models in BayesFlow

import os
os.environ["KERAS_BACKEND"] = "tensorflow"
import keras
import bayesflow as bf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import stats

def evidence_accumulation(nu, max_t, dt):
    timesteps = int(max_t / dt)
    t = np.linspace(0, max_t, timesteps)

    noise = np.random.normal(0, 1, size=timesteps) * np.sqrt(dt)
    evidence = nu * t + np.cumsum(noise)

    return t, evidence

t, evidence = evidence_accumulation(nu=2.5, max_t=1.0, dt=0.01)

plt.plot(t, evidence)
plt.xlabel("Time (s)")
plt.ylabel("Evidence")

Text(0, 0.5, 'Evidence')

Simple response time (Wald model)

Here we will train a simple response time model based on a single accumulator that diffuses with a drift \(\nu\) to a decision threshold \(\alpha\). For more background about the Wald model, see Anders, Alario, & van Maanen (2016). Here we will estimate only the drift and decision threshold, no non-decision time or variability in starting point. In this case, the mean and a standard deviation of the response times are sufficient statistics, so we do not need a summary network.

def prior():
    # drift rate
    nu=np.random.gamma(shape=10, scale=0.25)
    # decision threshold
    alpha=np.random.gamma(shape=10, scale=0.1)

    return dict(nu=nu, alpha=alpha)

# generate data for a single trial
def trial(nu, alpha, max_t, dt):
    t, evidence = evidence_accumulation(nu, max_t, dt)
    passage = np.argmax(evidence > alpha)
    rt = max_t if passage==0 else t[passage]

    return rt

# generate data for n trials
def likelihood(nu, alpha, n=250, max_t=3.0, dt=0.02):
    rt = np.zeros(n)
    for i in range(n):
        rt[i] = trial(nu, alpha, max_t, dt)

    return dict(rt=rt)

# sufficient statistics: mean, sd, n
def summary(rt):
    return dict(
        mean = np.mean(rt),
        sd = np.std(rt)
    )

simulator = bf.make_simulator([prior, likelihood, summary])

df = simulator.sample(1_000)

f=bf.diagnostics.pairs_samples(
    df, 
    variable_keys=["nu", "alpha", "mean", "sd"],
    variable_names=[r"$\nu$", r"$\alpha$", "mean RT", "sd RT"])

adapter = (bf.Adapter()
    .constrain(["nu", "alpha"], lower=0)
    .concatenate(["nu", "alpha"], into="inference_variables")
    .concatenate(["mean", "sd"], into="inference_conditions")
    .drop("rt")
)

workflow = bf.BasicWorkflow(
    simulator = simulator,
    adapter = adapter,
    inference_network = bf.networks.CouplingFlow(permutation="swap", subnet_kwargs=dict(dropout=False)),
    inference_variables = ["nu", "alpha"],
    inference_conditions = ["mean", "sd"]
)

train_data = simulator.sample(5_000)
validation_data = simulator.sample(1_000)

history=workflow.fit_offline(
    data=train_data, 
    epochs=50, 
    batch_size=250, 
    validation_data=validation_data
)

INFO:bayesflow:Fitting on dataset instance of OfflineDataset.
INFO:bayesflow:Building on a test batch.

Epoch 1/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 6s 31ms/step - loss: 4.3237 - loss/inference_loss: 4.3237 - val_loss: 3.2620 - val_loss/inference_loss: 3.2620

Epoch 2/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - loss: 2.9751 - loss/inference_loss: 2.9751 - val_loss: 2.6640 - val_loss/inference_loss: 2.6640

Epoch 3/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - loss: 2.5618 - loss/inference_loss: 2.5618 - val_loss: 2.4644 - val_loss/inference_loss: 2.4644

Epoch 4/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 10ms/step - loss: 2.3460 - loss/inference_loss: 2.3460 - val_loss: 2.2887 - val_loss/inference_loss: 2.2887

Epoch 5/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 10ms/step - loss: 2.1960 - loss/inference_loss: 2.1960 - val_loss: 2.0773 - val_loss/inference_loss: 2.0773

Epoch 6/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 2.0349 - loss/inference_loss: 2.0349 - val_loss: 1.9428 - val_loss/inference_loss: 1.9428

Epoch 7/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 1.8103 - loss/inference_loss: 1.8103 - val_loss: 1.6752 - val_loss/inference_loss: 1.6752

Epoch 8/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 1.4347 - loss/inference_loss: 1.4347 - val_loss: 1.0306 - val_loss/inference_loss: 1.0306

Epoch 9/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: 0.6755 - loss/inference_loss: 0.6755 - val_loss: 0.1332 - val_loss/inference_loss: 0.1332

Epoch 10/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -0.0722 - loss/inference_loss: -0.0722 - val_loss: -0.5113 - val_loss/inference_loss: -0.5113

Epoch 11/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -0.6561 - loss/inference_loss: -0.6561 - val_loss: -0.9005 - val_loss/inference_loss: -0.9005

Epoch 12/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -0.9802 - loss/inference_loss: -0.9802 - val_loss: -1.0020 - val_loss/inference_loss: -1.0020

Epoch 13/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -1.0407 - loss/inference_loss: -1.0407 - val_loss: -1.1719 - val_loss/inference_loss: -1.1719

Epoch 14/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -1.0787 - loss/inference_loss: -1.0787 - val_loss: -1.0381 - val_loss/inference_loss: -1.0381

Epoch 15/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: -1.0662 - loss/inference_loss: -1.0662 - val_loss: -1.1802 - val_loss/inference_loss: -1.1802

Epoch 16/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -1.1526 - loss/inference_loss: -1.1526 - val_loss: -1.1777 - val_loss/inference_loss: -1.1777

Epoch 17/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -1.1973 - loss/inference_loss: -1.1973 - val_loss: -1.3017 - val_loss/inference_loss: -1.3017

Epoch 18/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 12ms/step - loss: -1.2158 - loss/inference_loss: -1.2158 - val_loss: -1.1972 - val_loss/inference_loss: -1.1972

Epoch 19/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.2340 - loss/inference_loss: -1.2340 - val_loss: -1.1711 - val_loss/inference_loss: -1.1711

Epoch 20/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.2433 - loss/inference_loss: -1.2433 - val_loss: -1.3729 - val_loss/inference_loss: -1.3729

Epoch 21/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.2589 - loss/inference_loss: -1.2589 - val_loss: -1.2672 - val_loss/inference_loss: -1.2672

Epoch 22/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.2627 - loss/inference_loss: -1.2627 - val_loss: -1.1731 - val_loss/inference_loss: -1.1731

Epoch 23/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.2662 - loss/inference_loss: -1.2662 - val_loss: -1.2593 - val_loss/inference_loss: -1.2593

Epoch 24/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 18ms/step - loss: -1.2929 - loss/inference_loss: -1.2929 - val_loss: -1.3146 - val_loss/inference_loss: -1.3146

Epoch 25/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.2873 - loss/inference_loss: -1.2873 - val_loss: -1.1808 - val_loss/inference_loss: -1.1808

Epoch 26/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 13ms/step - loss: -1.2965 - loss/inference_loss: -1.2965 - val_loss: -1.2206 - val_loss/inference_loss: -1.2206

Epoch 27/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 17ms/step - loss: -1.2928 - loss/inference_loss: -1.2928 - val_loss: -1.2815 - val_loss/inference_loss: -1.2815

Epoch 28/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.3085 - loss/inference_loss: -1.3085 - val_loss: -1.2836 - val_loss/inference_loss: -1.2836

Epoch 29/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.3052 - loss/inference_loss: -1.3052 - val_loss: -1.3027 - val_loss/inference_loss: -1.3027

Epoch 30/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.3032 - loss/inference_loss: -1.3032 - val_loss: -1.3187 - val_loss/inference_loss: -1.3187

Epoch 31/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3289 - loss/inference_loss: -1.3289 - val_loss: -1.3867 - val_loss/inference_loss: -1.3867

Epoch 32/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.3229 - loss/inference_loss: -1.3229 - val_loss: -1.3211 - val_loss/inference_loss: -1.3211

Epoch 33/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3265 - loss/inference_loss: -1.3265 - val_loss: -1.3229 - val_loss/inference_loss: -1.3229

Epoch 34/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3420 - loss/inference_loss: -1.3420 - val_loss: -1.4363 - val_loss/inference_loss: -1.4363

Epoch 35/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3360 - loss/inference_loss: -1.3360 - val_loss: -1.3724 - val_loss/inference_loss: -1.3724

Epoch 36/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 14ms/step - loss: -1.3250 - loss/inference_loss: -1.3250 - val_loss: -1.4013 - val_loss/inference_loss: -1.4013

Epoch 37/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3528 - loss/inference_loss: -1.3528 - val_loss: -1.4209 - val_loss/inference_loss: -1.4209

Epoch 38/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3484 - loss/inference_loss: -1.3484 - val_loss: -1.3968 - val_loss/inference_loss: -1.3968

Epoch 39/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3481 - loss/inference_loss: -1.3481 - val_loss: -1.3297 - val_loss/inference_loss: -1.3297

Epoch 40/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3504 - loss/inference_loss: -1.3504 - val_loss: -1.4910 - val_loss/inference_loss: -1.4910

Epoch 41/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3585 - loss/inference_loss: -1.3585 - val_loss: -1.2492 - val_loss/inference_loss: -1.2492

Epoch 42/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 17ms/step - loss: -1.3609 - loss/inference_loss: -1.3609 - val_loss: -1.3917 - val_loss/inference_loss: -1.3917

Epoch 43/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3535 - loss/inference_loss: -1.3535 - val_loss: -1.3360 - val_loss/inference_loss: -1.3360

Epoch 44/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3583 - loss/inference_loss: -1.3583 - val_loss: -1.3302 - val_loss/inference_loss: -1.3302

Epoch 45/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3607 - loss/inference_loss: -1.3607 - val_loss: -1.3974 - val_loss/inference_loss: -1.3974

Epoch 46/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3665 - loss/inference_loss: -1.3665 - val_loss: -1.3183 - val_loss/inference_loss: -1.3183

Epoch 47/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3582 - loss/inference_loss: -1.3582 - val_loss: -1.2993 - val_loss/inference_loss: -1.2993

Epoch 48/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: -1.3658 - loss/inference_loss: -1.3658 - val_loss: -1.2053 - val_loss/inference_loss: -1.2053

Epoch 49/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 17ms/step - loss: -1.3575 - loss/inference_loss: -1.3575 - val_loss: -1.3078 - val_loss/inference_loss: -1.3078

Epoch 50/50

20/20 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: -1.3586 - loss/inference_loss: -1.3586 - val_loss: -1.3565 - val_loss/inference_loss: -1.3565

test_data = simulator.sample(1_000)
plots=workflow.plot_default_diagnostics(test_data=test_data)

Two choice task (Racing diffusion model)

We’ll assume a simple RDM with two choice alternatives (Tillman et al., 2020).

Here we will simplify the model to include no bias, no variability in starting points. Instead of modeling one accumulator for each of [left, right] responses, we will simply model one accumulator for “incorrect” and one accumulator for “correct” response. This makes it a bit easier to simulate from (we do not need to simulate stimuli).

The model has four parameters: 2 drift rates (incorrect - \(\nu_0\), correct \(\nu_1\)), decision threshold \(\alpha\), and non-decision time \(\tau\).

def context(n=None):
    if n is None:
        n = np.random.randint(200, 351)
    return dict(n=n)

def prior(nu=None, alpha=None, tau=None):
    if nu is None:
        nu=np.random.dirichlet([2, 2])
        nu=np.random.gamma(shape=5, scale=0.5) * nu
    if alpha is None:
        alpha=np.random.gamma(shape=5, scale=0.2)
    if tau is None:
        tau=np.random.exponential(0.15)
    else:
        tau=tau.item()

    return dict(nu=nu, alpha=alpha, tau=tau)

# generate data for a single trial
def trial(nu, alpha, tau, max_t, dt):
    response = -1
    min_t = max_t
    # loop over accumulators
    # if an accumulator has a smaller passage time than the current minimum
    # save it as the fastest accumulator (response)
    for resp, drift in enumerate(nu):
        t, evidence = evidence_accumulation(drift, max_t, dt)
        passage = np.argmax(evidence > alpha)
        t = max_t if passage==0 else t[passage]
        
        if t < min_t:
            min_t = t
            response = resp
            
    return min_t+tau, response

# generate data for n trials
# keep the data.shape always to max_n
# the rest is filled with 0s
def likelihood(n, nu, alpha, tau, max_t=3.0, dt=0.02, max_n=350):
    rt = np.zeros(max_n)
    response = np.zeros(max_n)
    observed = np.zeros(max_n)
    for i in range(n):
        result = trial(nu, alpha, tau, max_t, dt)
        rt[i] = result[0]
        response[i] = result[1]
        observed[i] = 1

    return dict(rt=rt, response=response, observed=observed)

simulator = bf.make_simulator([context, prior, likelihood])

adapter = (bf.Adapter()
    .as_set(["rt", "response", "observed"])
    .constrain(["nu", "alpha", "tau"], lower=0)
    .standardize(include="nu",    mean= 0.7, std=1.2)
    .standardize(include="alpha", mean= 0.5, std=0.7)
    .standardize(include="tau",   mean=-2.5, std=1.3)
    .concatenate(["nu", "alpha", "tau"], into="inference_variables")
    .concatenate(["rt", "response", "observed"], into="summary_variables")
    .rename("n", "inference_conditions")
)

workflow = bf.BasicWorkflow(
    simulator = simulator,
    adapter = adapter,
    inference_network = bf.networks.CouplingFlow(
        permutation="swap", 
        subnet_kwargs=dict(dropout=False)
    ),
    summary_network=bf.networks.DeepSet(
        base_distribution="normal",
        dropout=False
    ),
    inference_variables = ["nu", "alpha", "tau"],
    inference_conditions = ["n"],
    summary_variables = ["rt", "response", "observed"]
)

train_data = simulator.sample(5_000)
validation_data = simulator.sample(1_000)

history=workflow.fit_offline(
    data=train_data, 
    epochs=100, 
    batch_size=250, 
    validation_data=validation_data
)

INFO:bayesflow:Fitting on dataset instance of OfflineDataset.
INFO:bayesflow:Building on a test batch.

Epoch 1/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 34s 1s/step - loss: 14.3609 - loss/inference_loss: 14.0609 - loss/summary_loss: 0.3000 - val_loss: 6.1114 - val_loss/inference_loss: 5.9787 - val_loss/summary_loss: 0.1327

Epoch 2/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 633ms/step - loss: 5.8206 - loss/inference_loss: 5.6786 - loss/summary_loss: 0.1419 - val_loss: 5.5208 - val_loss/inference_loss: 5.3216 - val_loss/summary_loss: 0.1992

Epoch 3/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 645ms/step - loss: 5.2590 - loss/inference_loss: 5.0502 - loss/summary_loss: 0.2088 - val_loss: 5.1275 - val_loss/inference_loss: 4.9059 - val_loss/summary_loss: 0.2216

Epoch 4/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 649ms/step - loss: 4.7636 - loss/inference_loss: 4.5833 - loss/summary_loss: 0.1803 - val_loss: 4.5547 - val_loss/inference_loss: 4.4057 - val_loss/summary_loss: 0.1490

Epoch 5/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 607ms/step - loss: 4.4397 - loss/inference_loss: 4.2925 - loss/summary_loss: 0.1471 - val_loss: 4.3656 - val_loss/inference_loss: 4.2313 - val_loss/summary_loss: 0.1344

Epoch 6/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 604ms/step - loss: 4.3418 - loss/inference_loss: 4.1914 - loss/summary_loss: 0.1504 - val_loss: 4.1845 - val_loss/inference_loss: 4.0441 - val_loss/summary_loss: 0.1405

Epoch 7/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 605ms/step - loss: 4.3237 - loss/inference_loss: 4.1712 - loss/summary_loss: 0.1525 - val_loss: 4.3240 - val_loss/inference_loss: 4.1473 - val_loss/summary_loss: 0.1767

Epoch 8/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 613ms/step - loss: 4.1682 - loss/inference_loss: 4.0072 - loss/summary_loss: 0.1610 - val_loss: 3.9488 - val_loss/inference_loss: 3.8090 - val_loss/summary_loss: 0.1398

Epoch 9/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 607ms/step - loss: 3.9366 - loss/inference_loss: 3.7888 - loss/summary_loss: 0.1479 - val_loss: 4.2893 - val_loss/inference_loss: 4.1522 - val_loss/summary_loss: 0.1371

Epoch 10/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 607ms/step - loss: 3.8633 - loss/inference_loss: 3.7098 - loss/summary_loss: 0.1535 - val_loss: 4.0335 - val_loss/inference_loss: 3.9091 - val_loss/summary_loss: 0.1245

Epoch 11/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 607ms/step - loss: 3.7262 - loss/inference_loss: 3.5701 - loss/summary_loss: 0.1562 - val_loss: 3.7161 - val_loss/inference_loss: 3.5764 - val_loss/summary_loss: 0.1397

Epoch 12/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 609ms/step - loss: 3.6239 - loss/inference_loss: 3.4630 - loss/summary_loss: 0.1610 - val_loss: 3.5772 - val_loss/inference_loss: 3.4350 - val_loss/summary_loss: 0.1422

Epoch 13/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 622ms/step - loss: 3.4660 - loss/inference_loss: 3.2979 - loss/summary_loss: 0.1681 - val_loss: 3.2676 - val_loss/inference_loss: 3.1305 - val_loss/summary_loss: 0.1371

Epoch 14/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 622ms/step - loss: 3.4642 - loss/inference_loss: 3.3057 - loss/summary_loss: 0.1586 - val_loss: 3.6891 - val_loss/inference_loss: 3.5483 - val_loss/summary_loss: 0.1408

Epoch 15/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 612ms/step - loss: 3.3781 - loss/inference_loss: 3.2192 - loss/summary_loss: 0.1588 - val_loss: 3.4303 - val_loss/inference_loss: 3.2832 - val_loss/summary_loss: 0.1471

Epoch 16/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 607ms/step - loss: 3.1897 - loss/inference_loss: 3.0271 - loss/summary_loss: 0.1626 - val_loss: 3.6857 - val_loss/inference_loss: 3.5336 - val_loss/summary_loss: 0.1521

Epoch 17/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 611ms/step - loss: 3.1959 - loss/inference_loss: 3.0351 - loss/summary_loss: 0.1608 - val_loss: 3.4023 - val_loss/inference_loss: 3.2598 - val_loss/summary_loss: 0.1426

Epoch 18/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 624ms/step - loss: 3.0804 - loss/inference_loss: 2.9221 - loss/summary_loss: 0.1583 - val_loss: 2.8908 - val_loss/inference_loss: 2.7297 - val_loss/summary_loss: 0.1611

Epoch 19/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 618ms/step - loss: 3.0177 - loss/inference_loss: 2.8528 - loss/summary_loss: 0.1649 - val_loss: 3.3557 - val_loss/inference_loss: 3.1970 - val_loss/summary_loss: 0.1587

Epoch 20/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 620ms/step - loss: 2.9631 - loss/inference_loss: 2.7981 - loss/summary_loss: 0.1650 - val_loss: 3.7985 - val_loss/inference_loss: 3.6307 - val_loss/summary_loss: 0.1678

Epoch 21/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 606ms/step - loss: 2.9434 - loss/inference_loss: 2.7783 - loss/summary_loss: 0.1652 - val_loss: 2.8577 - val_loss/inference_loss: 2.7084 - val_loss/summary_loss: 0.1493

Epoch 22/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 12s 618ms/step - loss: 2.9067 - loss/inference_loss: 2.7405 - loss/summary_loss: 0.1662 - val_loss: 3.5980 - val_loss/inference_loss: 3.4340 - val_loss/summary_loss: 0.1640

Epoch 23/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 664ms/step - loss: 2.9271 - loss/inference_loss: 2.7585 - loss/summary_loss: 0.1686 - val_loss: 3.5134 - val_loss/inference_loss: 3.3457 - val_loss/summary_loss: 0.1677

Epoch 24/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 666ms/step - loss: 2.8916 - loss/inference_loss: 2.7192 - loss/summary_loss: 0.1724 - val_loss: 2.5384 - val_loss/inference_loss: 2.3735 - val_loss/summary_loss: 0.1649

Epoch 25/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 655ms/step - loss: 2.6834 - loss/inference_loss: 2.5162 - loss/summary_loss: 0.1673 - val_loss: 2.9493 - val_loss/inference_loss: 2.7742 - val_loss/summary_loss: 0.1752

Epoch 26/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 663ms/step - loss: 2.7069 - loss/inference_loss: 2.5339 - loss/summary_loss: 0.1729 - val_loss: 2.7834 - val_loss/inference_loss: 2.6410 - val_loss/summary_loss: 0.1424

Epoch 27/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 631ms/step - loss: 2.7247 - loss/inference_loss: 2.5504 - loss/summary_loss: 0.1743 - val_loss: 2.7583 - val_loss/inference_loss: 2.6091 - val_loss/summary_loss: 0.1491

Epoch 28/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 655ms/step - loss: 3.0438 - loss/inference_loss: 2.8668 - loss/summary_loss: 0.1770 - val_loss: 2.7321 - val_loss/inference_loss: 2.5490 - val_loss/summary_loss: 0.1831

Epoch 29/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 646ms/step - loss: 3.3625 - loss/inference_loss: 3.1734 - loss/summary_loss: 0.1891 - val_loss: 3.2479 - val_loss/inference_loss: 3.0610 - val_loss/summary_loss: 0.1869

Epoch 30/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 2.8979 - loss/inference_loss: 2.7022 - loss/summary_loss: 0.1956 - val_loss: 3.0476 - val_loss/inference_loss: 2.8501 - val_loss/summary_loss: 0.1975

Epoch 31/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 642ms/step - loss: 2.8198 - loss/inference_loss: 2.6258 - loss/summary_loss: 0.1939 - val_loss: 2.5580 - val_loss/inference_loss: 2.3881 - val_loss/summary_loss: 0.1699

Epoch 32/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 652ms/step - loss: 2.6621 - loss/inference_loss: 2.4773 - loss/summary_loss: 0.1847 - val_loss: 2.7204 - val_loss/inference_loss: 2.5579 - val_loss/summary_loss: 0.1626

Epoch 33/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 641ms/step - loss: 2.3376 - loss/inference_loss: 2.1612 - loss/summary_loss: 0.1764 - val_loss: 2.3137 - val_loss/inference_loss: 2.1466 - val_loss/summary_loss: 0.1671

Epoch 34/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 640ms/step - loss: 2.3064 - loss/inference_loss: 2.1392 - loss/summary_loss: 0.1671 - val_loss: 2.1724 - val_loss/inference_loss: 2.0077 - val_loss/summary_loss: 0.1647

Epoch 35/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 2.6591 - loss/inference_loss: 2.4902 - loss/summary_loss: 0.1689 - val_loss: 2.8983 - val_loss/inference_loss: 2.7284 - val_loss/summary_loss: 0.1699

Epoch 36/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 2.2897 - loss/inference_loss: 2.0997 - loss/summary_loss: 0.1900 - val_loss: 2.1468 - val_loss/inference_loss: 1.9434 - val_loss/summary_loss: 0.2034

Epoch 37/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 672ms/step - loss: 2.3447 - loss/inference_loss: 2.1427 - loss/summary_loss: 0.2020 - val_loss: 2.2293 - val_loss/inference_loss: 2.0126 - val_loss/summary_loss: 0.2167

Epoch 38/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 650ms/step - loss: 2.1639 - loss/inference_loss: 1.9669 - loss/summary_loss: 0.1970 - val_loss: 2.2016 - val_loss/inference_loss: 2.0265 - val_loss/summary_loss: 0.1751

Epoch 39/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 631ms/step - loss: 1.9992 - loss/inference_loss: 1.8057 - loss/summary_loss: 0.1936 - val_loss: 1.9904 - val_loss/inference_loss: 1.8133 - val_loss/summary_loss: 0.1771

Epoch 40/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 637ms/step - loss: 1.8934 - loss/inference_loss: 1.7005 - loss/summary_loss: 0.1929 - val_loss: 2.1794 - val_loss/inference_loss: 2.0040 - val_loss/summary_loss: 0.1755

Epoch 41/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 1.8562 - loss/inference_loss: 1.6690 - loss/summary_loss: 0.1872 - val_loss: 1.9737 - val_loss/inference_loss: 1.7985 - val_loss/summary_loss: 0.1752

Epoch 42/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 632ms/step - loss: 1.7466 - loss/inference_loss: 1.5612 - loss/summary_loss: 0.1854 - val_loss: 2.0639 - val_loss/inference_loss: 1.8752 - val_loss/summary_loss: 0.1887

Epoch 43/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 648ms/step - loss: 1.7693 - loss/inference_loss: 1.5840 - loss/summary_loss: 0.1853 - val_loss: 2.5070 - val_loss/inference_loss: 2.3081 - val_loss/summary_loss: 0.1989

Epoch 44/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 674ms/step - loss: 1.7723 - loss/inference_loss: 1.5713 - loss/summary_loss: 0.2011 - val_loss: 1.9744 - val_loss/inference_loss: 1.7753 - val_loss/summary_loss: 0.1991

Epoch 45/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 634ms/step - loss: 1.6406 - loss/inference_loss: 1.4498 - loss/summary_loss: 0.1909 - val_loss: 1.7236 - val_loss/inference_loss: 1.5469 - val_loss/summary_loss: 0.1768

Epoch 46/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 1.6229 - loss/inference_loss: 1.4347 - loss/summary_loss: 0.1882 - val_loss: 1.7695 - val_loss/inference_loss: 1.5858 - val_loss/summary_loss: 0.1838

Epoch 47/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 635ms/step - loss: 1.9651 - loss/inference_loss: 1.7786 - loss/summary_loss: 0.1865 - val_loss: 2.2095 - val_loss/inference_loss: 2.0418 - val_loss/summary_loss: 0.1676

Epoch 48/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 1.7954 - loss/inference_loss: 1.6079 - loss/summary_loss: 0.1875 - val_loss: 1.7250 - val_loss/inference_loss: 1.5349 - val_loss/summary_loss: 0.1901

Epoch 49/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 640ms/step - loss: 1.4650 - loss/inference_loss: 1.2654 - loss/summary_loss: 0.1996 - val_loss: 1.5375 - val_loss/inference_loss: 1.3673 - val_loss/summary_loss: 0.1703

Epoch 50/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 1.4561 - loss/inference_loss: 1.2723 - loss/summary_loss: 0.1838 - val_loss: 1.3710 - val_loss/inference_loss: 1.2069 - val_loss/summary_loss: 0.1641

Epoch 51/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 652ms/step - loss: 1.3965 - loss/inference_loss: 1.2208 - loss/summary_loss: 0.1757 - val_loss: 1.6446 - val_loss/inference_loss: 1.4805 - val_loss/summary_loss: 0.1641

Epoch 52/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 647ms/step - loss: 1.5803 - loss/inference_loss: 1.4120 - loss/summary_loss: 0.1683 - val_loss: 2.0261 - val_loss/inference_loss: 1.8793 - val_loss/summary_loss: 0.1468

Epoch 53/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 646ms/step - loss: 1.4886 - loss/inference_loss: 1.3165 - loss/summary_loss: 0.1721 - val_loss: 1.4217 - val_loss/inference_loss: 1.2650 - val_loss/summary_loss: 0.1567

Epoch 54/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 649ms/step - loss: 1.3557 - loss/inference_loss: 1.1763 - loss/summary_loss: 0.1794 - val_loss: 1.1852 - val_loss/inference_loss: 1.0335 - val_loss/summary_loss: 0.1517

Epoch 55/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 647ms/step - loss: 1.3222 - loss/inference_loss: 1.1358 - loss/summary_loss: 0.1863 - val_loss: 1.4570 - val_loss/inference_loss: 1.2818 - val_loss/summary_loss: 0.1752

Epoch 56/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 651ms/step - loss: 1.2665 - loss/inference_loss: 1.0845 - loss/summary_loss: 0.1819 - val_loss: 1.3704 - val_loss/inference_loss: 1.2134 - val_loss/summary_loss: 0.1570

Epoch 57/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 650ms/step - loss: 1.1603 - loss/inference_loss: 0.9831 - loss/summary_loss: 0.1771 - val_loss: 1.2693 - val_loss/inference_loss: 1.1043 - val_loss/summary_loss: 0.1650

Epoch 58/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 667ms/step - loss: 1.1930 - loss/inference_loss: 1.0246 - loss/summary_loss: 0.1684 - val_loss: 1.1312 - val_loss/inference_loss: 0.9607 - val_loss/summary_loss: 0.1705

Epoch 59/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 646ms/step - loss: 1.3012 - loss/inference_loss: 1.1337 - loss/summary_loss: 0.1675 - val_loss: 0.9646 - val_loss/inference_loss: 0.8219 - val_loss/summary_loss: 0.1427

Epoch 60/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 1.1655 - loss/inference_loss: 0.9962 - loss/summary_loss: 0.1692 - val_loss: 1.2670 - val_loss/inference_loss: 1.1023 - val_loss/summary_loss: 0.1647

Epoch 61/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 642ms/step - loss: 1.0884 - loss/inference_loss: 0.9162 - loss/summary_loss: 0.1722 - val_loss: 1.0760 - val_loss/inference_loss: 0.9093 - val_loss/summary_loss: 0.1666

Epoch 62/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 1.0608 - loss/inference_loss: 0.8926 - loss/summary_loss: 0.1682 - val_loss: 1.2358 - val_loss/inference_loss: 1.0848 - val_loss/summary_loss: 0.1510

Epoch 63/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 645ms/step - loss: 0.9816 - loss/inference_loss: 0.8134 - loss/summary_loss: 0.1682 - val_loss: 1.1988 - val_loss/inference_loss: 1.0484 - val_loss/summary_loss: 0.1503

Epoch 64/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 652ms/step - loss: 0.9914 - loss/inference_loss: 0.8276 - loss/summary_loss: 0.1638 - val_loss: 1.1647 - val_loss/inference_loss: 1.0099 - val_loss/summary_loss: 0.1549

Epoch 65/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 663ms/step - loss: 0.9703 - loss/inference_loss: 0.8106 - loss/summary_loss: 0.1597 - val_loss: 1.0473 - val_loss/inference_loss: 0.8970 - val_loss/summary_loss: 0.1503

Epoch 66/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 649ms/step - loss: 0.9324 - loss/inference_loss: 0.7728 - loss/summary_loss: 0.1596 - val_loss: 1.1476 - val_loss/inference_loss: 0.9971 - val_loss/summary_loss: 0.1505

Epoch 67/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 646ms/step - loss: 0.9491 - loss/inference_loss: 0.7905 - loss/summary_loss: 0.1585 - val_loss: 0.9906 - val_loss/inference_loss: 0.8568 - val_loss/summary_loss: 0.1338

Epoch 68/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 645ms/step - loss: 0.9203 - loss/inference_loss: 0.7623 - loss/summary_loss: 0.1580 - val_loss: 0.8555 - val_loss/inference_loss: 0.7052 - val_loss/summary_loss: 0.1503

Epoch 69/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 647ms/step - loss: 0.9002 - loss/inference_loss: 0.7434 - loss/summary_loss: 0.1567 - val_loss: 0.9529 - val_loss/inference_loss: 0.7949 - val_loss/summary_loss: 0.1580

Epoch 70/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 642ms/step - loss: 0.8696 - loss/inference_loss: 0.7143 - loss/summary_loss: 0.1553 - val_loss: 1.0420 - val_loss/inference_loss: 0.8976 - val_loss/summary_loss: 0.1444

Epoch 71/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 635ms/step - loss: 0.8365 - loss/inference_loss: 0.6796 - loss/summary_loss: 0.1569 - val_loss: 0.7756 - val_loss/inference_loss: 0.6244 - val_loss/summary_loss: 0.1512

Epoch 72/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 641ms/step - loss: 0.7884 - loss/inference_loss: 0.6326 - loss/summary_loss: 0.1558 - val_loss: 0.9432 - val_loss/inference_loss: 0.8026 - val_loss/summary_loss: 0.1406

Epoch 73/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 654ms/step - loss: 0.8110 - loss/inference_loss: 0.6554 - loss/summary_loss: 0.1556 - val_loss: 1.2307 - val_loss/inference_loss: 1.0674 - val_loss/summary_loss: 0.1633

Epoch 74/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 645ms/step - loss: 0.8237 - loss/inference_loss: 0.6710 - loss/summary_loss: 0.1527 - val_loss: 1.0769 - val_loss/inference_loss: 0.9334 - val_loss/summary_loss: 0.1435

Epoch 75/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 640ms/step - loss: 0.7520 - loss/inference_loss: 0.6001 - loss/summary_loss: 0.1519 - val_loss: 0.5952 - val_loss/inference_loss: 0.4440 - val_loss/summary_loss: 0.1512

Epoch 76/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 638ms/step - loss: 0.7704 - loss/inference_loss: 0.6198 - loss/summary_loss: 0.1506 - val_loss: 1.0217 - val_loss/inference_loss: 0.8737 - val_loss/summary_loss: 0.1481

Epoch 77/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 633ms/step - loss: 0.8600 - loss/inference_loss: 0.7117 - loss/summary_loss: 0.1483 - val_loss: 0.8531 - val_loss/inference_loss: 0.7029 - val_loss/summary_loss: 0.1502

Epoch 78/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 646ms/step - loss: 0.7354 - loss/inference_loss: 0.5853 - loss/summary_loss: 0.1500 - val_loss: 0.8464 - val_loss/inference_loss: 0.6999 - val_loss/summary_loss: 0.1465

Epoch 79/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 656ms/step - loss: 0.6935 - loss/inference_loss: 0.5461 - loss/summary_loss: 0.1474 - val_loss: 1.1493 - val_loss/inference_loss: 1.0016 - val_loss/summary_loss: 0.1477

Epoch 80/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 640ms/step - loss: 0.6776 - loss/inference_loss: 0.5293 - loss/summary_loss: 0.1483 - val_loss: 0.8535 - val_loss/inference_loss: 0.7131 - val_loss/summary_loss: 0.1404

Epoch 81/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 641ms/step - loss: 0.6657 - loss/inference_loss: 0.5185 - loss/summary_loss: 0.1471 - val_loss: 0.6595 - val_loss/inference_loss: 0.5300 - val_loss/summary_loss: 0.1295

Epoch 82/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 655ms/step - loss: 0.6525 - loss/inference_loss: 0.5066 - loss/summary_loss: 0.1459 - val_loss: 0.8119 - val_loss/inference_loss: 0.6816 - val_loss/summary_loss: 0.1303

Epoch 83/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 652ms/step - loss: 0.6473 - loss/inference_loss: 0.5016 - loss/summary_loss: 0.1457 - val_loss: 1.1176 - val_loss/inference_loss: 0.9759 - val_loss/summary_loss: 0.1417

Epoch 84/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 651ms/step - loss: 0.6252 - loss/inference_loss: 0.4787 - loss/summary_loss: 0.1464 - val_loss: 0.9897 - val_loss/inference_loss: 0.8482 - val_loss/summary_loss: 0.1415

Epoch 85/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 648ms/step - loss: 0.6253 - loss/inference_loss: 0.4791 - loss/summary_loss: 0.1461 - val_loss: 0.9309 - val_loss/inference_loss: 0.7894 - val_loss/summary_loss: 0.1415

Epoch 86/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 653ms/step - loss: 0.6400 - loss/inference_loss: 0.4954 - loss/summary_loss: 0.1446 - val_loss: 0.7634 - val_loss/inference_loss: 0.6060 - val_loss/summary_loss: 0.1574

Epoch 87/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 634ms/step - loss: 0.6005 - loss/inference_loss: 0.4567 - loss/summary_loss: 0.1438 - val_loss: 0.9929 - val_loss/inference_loss: 0.8591 - val_loss/summary_loss: 0.1337

Epoch 88/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 0.5821 - loss/inference_loss: 0.4381 - loss/summary_loss: 0.1440 - val_loss: 0.7725 - val_loss/inference_loss: 0.6350 - val_loss/summary_loss: 0.1375

Epoch 89/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 656ms/step - loss: 0.5848 - loss/inference_loss: 0.4408 - loss/summary_loss: 0.1440 - val_loss: 0.7279 - val_loss/inference_loss: 0.5971 - val_loss/summary_loss: 0.1309

Epoch 90/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 654ms/step - loss: 0.5676 - loss/inference_loss: 0.4243 - loss/summary_loss: 0.1433 - val_loss: 0.4774 - val_loss/inference_loss: 0.3479 - val_loss/summary_loss: 0.1295

Epoch 91/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 650ms/step - loss: 0.5667 - loss/inference_loss: 0.4229 - loss/summary_loss: 0.1438 - val_loss: 0.8972 - val_loss/inference_loss: 0.7705 - val_loss/summary_loss: 0.1267

Epoch 92/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 644ms/step - loss: 0.5633 - loss/inference_loss: 0.4198 - loss/summary_loss: 0.1435 - val_loss: 0.5660 - val_loss/inference_loss: 0.4268 - val_loss/summary_loss: 0.1392

Epoch 93/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 651ms/step - loss: 0.5625 - loss/inference_loss: 0.4200 - loss/summary_loss: 0.1425 - val_loss: 0.8851 - val_loss/inference_loss: 0.7444 - val_loss/summary_loss: 0.1407

Epoch 94/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 653ms/step - loss: 0.5464 - loss/inference_loss: 0.4034 - loss/summary_loss: 0.1429 - val_loss: 0.7550 - val_loss/inference_loss: 0.6080 - val_loss/summary_loss: 0.1470

Epoch 95/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 659ms/step - loss: 0.5542 - loss/inference_loss: 0.4115 - loss/summary_loss: 0.1427 - val_loss: 1.0188 - val_loss/inference_loss: 0.8732 - val_loss/summary_loss: 0.1456

Epoch 96/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 651ms/step - loss: 0.5547 - loss/inference_loss: 0.4116 - loss/summary_loss: 0.1431 - val_loss: 0.7003 - val_loss/inference_loss: 0.5620 - val_loss/summary_loss: 0.1383

Epoch 97/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 650ms/step - loss: 0.5643 - loss/inference_loss: 0.4210 - loss/summary_loss: 0.1432 - val_loss: 0.5191 - val_loss/inference_loss: 0.3852 - val_loss/summary_loss: 0.1339

Epoch 98/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 633ms/step - loss: 0.5524 - loss/inference_loss: 0.4096 - loss/summary_loss: 0.1428 - val_loss: 0.7010 - val_loss/inference_loss: 0.5694 - val_loss/summary_loss: 0.1316

Epoch 99/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 640ms/step - loss: 0.5481 - loss/inference_loss: 0.4046 - loss/summary_loss: 0.1436 - val_loss: 0.8737 - val_loss/inference_loss: 0.7477 - val_loss/summary_loss: 0.1261

Epoch 100/100

20/20 ━━━━━━━━━━━━━━━━━━━━ 13s 637ms/step - loss: 0.5406 - loss/inference_loss: 0.3977 - loss/summary_loss: 0.1429 - val_loss: 0.6171 - val_loss/inference_loss: 0.4793 - val_loss/summary_loss: 0.1378

test_data = simulator.sample(1_000)
plots=workflow.plot_default_diagnostics(test_data=test_data)

Application to real data

Here we will work with data from Fortmann, et al. (2008), that is also included in the R package EMC2 (Stevenson, Donzallaz, & Heathcote, 2025). Here we use a slightly reshaped dataset where each combination of subject x condition is extended to a length of 350 trials (the missing trials are filled with 0s). This allows us to estimate the posterior for each subject in each condition with a single pass through the posterior approximator.

data_inference = pd.read_csv("forstmann.csv")

data_inference['condition'].unique()

array(['accuracy', 'speed', 'neutral'], dtype=object)

data_inference_grouped = data_inference.groupby(["subject", "condition"])

data_inference_dict = {
    key: np.array([group[key].values.reshape(350, 1) for _, group in data_inference_grouped]) 
             for key in ['rt', 'response', 'observed']}
data_inference_dict["n"] = np.sum(data_inference_dict["observed"], axis=1)
print({key: value.shape for key, value in data_inference_dict.items()})

{'rt': (57, 350, 1), 'response': (57, 350, 1), 'observed': (57, 350, 1), 'n': (57, 1)}

posterior_samples = workflow.sample(conditions=data_inference_dict, num_samples=1_000)

# pick the first participant, first condition
posterior = {key: value[0] for key, value in posterior_samples.items()}
data = data_inference_grouped.get_group(('as1t', 'accuracy'))

f=bf.diagnostics.pairs_posterior(estimates=posterior)

def ecdf(rt, response, observed, **kwargs):
    observed_mask = (observed == 1)
    response_0_mask = ((response == 0) & observed_mask)
    response_1_mask = ((response == 1) & observed_mask)

    response_0_prop = np.sum(response_0_mask) / np.sum(observed_mask)
    response_1_prop = np.sum(response_1_mask) / np.sum(observed_mask)

    response_0_ecdf = stats.ecdf(rt[response_0_mask]).cdf
    response_0_ecdf = response_0_prop * response_0_ecdf.evaluate(np.linspace(0, 1, 101))

    response_1_ecdf = stats.ecdf(rt[response_1_mask]).cdf
    response_1_ecdf = response_1_prop * response_1_ecdf.evaluate(np.linspace(0, 1, 101))

    return response_0_ecdf, response_1_ecdf

plot_data = ecdf(**data)

posterior_predictives = simulator.sample(1000, **posterior)

plot_data_predictive = []
for i in range(1000):
    x = { key: value[i:i+1,...] for key, value in posterior_predictives.items()}
    plot_data_predictive.append(ecdf(**x))
plot_data_predictive = np.array(plot_data_predictive)

plot_data_quantiles = np.quantile(
    plot_data_predictive, 
    q = [0.25, 0.5, 0.75],
    axis=0
)
plot_data_quantiles.shape

(3, 2, 101)

t = np.linspace(0, 1, 101)
cols = ["red", "blue"]
for i, lab in enumerate(["Incorrect", "Correct"]):
    plt.plot(t, plot_data[i], label=lab, color=cols[i])
    plt.plot(t, plot_data_quantiles[1, i, :], color=cols[i], alpha=0.5, label="median predictive")
    plt.fill_between(
        t,
        plot_data_quantiles[0,  i,:],
        plot_data_quantiles[-1, i,:],
        label="50% predictive interval",
        color=cols[i],
        alpha=0.3
    )
f=plt.legend()

Further exercises

The diagnostics show that there are some situations where we learn basically nothing from the data. Find out whether there is some common pattern between these simulations and whether you could fix this.
Each participant in the experiment was exposed to three different conditions (speed, neutral, accuracy). Here we fitted a simple model to each condition separately. However, it would make much more sense to fit a single model that encompassess all three conditions. Think about which parameters of the decision making model should theoretically be affected by the experimental condition, and implement a model that quantifies these differences.
The data we worked with here are recoded such that we only have correct vs incorrect responses. However, the actual data contain two columns, where the stimulus has values “left” vs “right” and the response as well, “left” or “right”. Think about how to reparametrize the model such that instead of having one accumulator for correct and one for incorrect responses, you would have one accumulator for “left” and one accumulator “right” – and how would you make sure that the model responds to changes in stimulus. Train a BayesFlow model based on data from such simulator.
The function that simulates the evidence accumulation process (evidence_accumulation) generates a standard Wiener diffusion with drift. Look for alternative ways how you could represent this process and try to implement it in a statistical model (examples: linear accumulation, Ornstein-Uhlenbeck process, Levy flight, …).

References

Anders, R., Alario, F., & Van Maanen, L. (2016). The shifted Wald distribution for response time data analysis. Psychological methods, 21(3), 309.

Forstmann, B. U., Dutilh, G., Brown, S., Neumann, J., Von Cramon, D. Y., Ridderinkhof, K. R., & Wagenmakers, E. J. (2008). Striatum and pre-SMA facilitate decision-making under time pressure. Proceedings of the National Academy of Sciences, 105(45), 17538-17542.

Stevenson N., Donzallaz M., Heathcote A. (2025). EMC2: Bayesian Hierarchical Analysis of Cognitive Models of Choice. R package version 3.1.0, https://github.com/ampl-psych/emc2.

Tillman, G., Van Zandt, T., & Logan, G. D. (2020). Sequential sampling models without random between-trial variability: The racing diffusion model of speeded decision making. Psychonomic Bulletin & Review, 27(5), 911-936.