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A Bayesian hierarchical model of trial-to-trial fluctuations in decision criterion

Fig 2

An overview of the generative model and its hierarchical structure.

A) The figure displays, using 100 simulated trials for one agent, responses sampled from a Bernoulli distribution (i.e., a series of weighted coin flips). The trial-by-trial Bernoulli probabilities are a function of a weighted combination of the observed covariates and the latent criterion fluctuations x_t. The first three columns show how the covariates, drawn from a standard normal distribution evolve randomly over time. Summing the weighted covariates with the criterion trajectory results in the log-odds of the Bernoulli distribution. After applying a sigmoid transformation to transform log-odds into probabilities, we draw from the Bernoulli distribution (i.e., weighted coin flip) to produce binary responses. B) For each subject i we infer weights () and model the latent criterion fluctuations x_i,t as an AR(1) process with per-subject intercept b_i, autoregressive coefficient a_i, and variance . Through the specification of hierarchical priors we allow the sharing of statistical strength across subjects when estimating these individual parameters. Like the individual parameters, the parameters of the hierarchical distributions are iteratively updated during the estimation procedure. Note that b_i is not estimated directly and therefore does not have a hierarchical prior. Instead, we estimate the mean of the criterion trajectory , for which a normal hierarchical prior with zero mean and as variance is assumed. Following the formula for the mean of an AR(1) process we can derive b_i = (1−a_i).

doi: https://doi.org/10.1371/journal.pcbi.1013291.g002