Model guide

The accumulator race

Each accumulator begins at a point drawn uniformly between zero and A. Its drift is drawn from a normal distribution with mean v and standard deviation s, independently truncated above zero. Evidence increases at that constant drift until it reaches threshold b.

The finish time is tau + (b - start) / drift. The first accumulator to finish determines both the response and its response time.

Parameters

Parameter	Meaning	Domain
A	Upper bound of the uniform starting point	Positive
b	Response threshold	Greater than A
v	Mean of the underlying, untruncated normal drift	Finite
s	Standard deviation of the underlying normal drift	Positive
tau	Non-decision time	Non-negative

Times and non-decision times must use the same units. v is the mean of the underlying normal distribution, not the mean after truncation.

Working with parameters

Parameter fields accept scalars or one-dimensional vectors. Scalars broadcast across the race; vectors must match its accumulator count.

from pyLBA import LBAParameters

parameters = LBAParameters(A=1, b=[2, 3], v=2, s=1, tau=0.1)
print(parameters.validate())
values = parameters.to_dict()
restored = LBAParameters.from_dict(values)
thresholds = parameters.b.value

validate checks numeric parameter values and shapes. Simulation rejects invalid parameters. During fitting, parameter objects can also carry symbolic PyTensor variables.

Data format

Input data is a nonempty pandas DataFrame with finite positive rt values and non-negative integer response labels. Choices are numbered from zero.

import pandas as pd

data = pd.DataFrame({
    "rt": [0.5, 0.7, 0.4, 0.9],
    "response": [0, 1, 0, 1],
})

Fitting infers the number of accumulators from the largest response label plus one. Pass n_acc when the highest-numbered choices have no observations.

Likelihood

LBAModel.pdf returns a PyTensor expression for the joint density of response time and choice. Each first-passage density and CDF is normalized by the probability of a positive drift. The winning density is multiplied by the survival probabilities of the other accumulators. This matches simulation’s independent positive truncation.

from pyLBA import LBAModel

model = LBAModel()
density = model.pdf(
    data.rt.to_numpy(), data.response.to_numpy(), parameters, n_acc=2
).eval()

The result is a density, not a log likelihood. It is zero outside the model’s parameter or decision-time support.

Reference

Brown, S. D., & Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive Psychology, 57(3), 153–178.