Bayesian inference with Stan: A tutorial on adding custom distributions

Abstract

When evaluating cognitive models based on fits to observed data (or, really, any model that has free parameters), parameter estimation is critically important. Traditional techniques like hill climbing by minimizing or maximizing a fit statistic often result in point estimates. Bayesian approaches instead estimate parameters as posterior probability distributions, and thus naturally account for the uncertainty associated with parameter estimation; Bayesian approaches also offer powerful and principled methods for model comparison. Although software applications such as WinBUGS (Lunn, Thomas, Best, & Spiegelhalter, Statistics and Computing, 10, 325–337, 2000) and JAGS (Plummer, 2003) provide “turnkey”-style packages for Bayesian inference, they can be inefficient when dealing with models whose parameters are correlated, which is often the case for cognitive models, and they can impose significant technical barriers to adding custom distributions, which is often necessary when implementing cognitive models within a Bayesian framework. A recently developed software package called Stan (Stan Development Team, 2015) can solve both problems, as well as provide a turnkey solution to Bayesian inference. We present a tutorial on how to use Stan and how to add custom distributions to it, with an example using the linear ballistic accumulator model (Brown & Heathcote, Cognitive Psychology, 57, 153–178. doi:10.1016/j.cogpsych.2007.12.002, 2008).

Bayesian statistical approaches to evaluating cognitive models

“…As such, interest in the Bayesian approach to evaluating cognitive models has grown significantly over the past few years. Some applications include evaluating variants of signal detection theory (e.g., Refs ), multinomial processing trees (e.g., Ref ), individual differences (e.g., Refs ), decision making (e.g., Refs ), multidimensional scaling (e.g., Refs ), choice response time (e.g., Refs ), memory (e.g., Refs ), and joint modeling of neural and behavioral data (e.g., Refs ).…”

“…For many, if not most, cognitive models, it is necessary to implement custom probability distributions. WinBUGS and JAGS allow this, but require relatively low‐level programming in C++; Stan allows this within the same Stan programming language directly …”

“…Throughout, we use a cognitive model called the Linear Ballistic Accumulator model (LBA; 54 ) as a running example. We chose the LBA because it is a general model of choice response time that can be applied to a wide variety of tasks; we also recently published a companion paper 45 that outlines how to implement the LBA in a Bayesian statistical language called Stan. 27 There are many excellent reviews and textbooks on Bayesian approaches to evaluating cognitive models and Bayesian statistics more generally.…”

Deployable probabilistic programming

“…To use a distribution in a differentiable model, the distribution's log probability density function must be differentiable by both the distribution's parameters and by the value of the random variable. Distributions are usually supplied as a part of the probabilistic programming framework [10,11,51], and, if at all possible, adding a user-defined distribution requires programming at a lower level of abstraction than while specifying a probabilistic model [1,41,51].…”

Modelling individual difference in visual categorization

“…Consider first a Bayesian hierarchical version of the diffusion model (e.g., Vandekerckhove, Tuerlinckx, & Lee, 2011) or of the linear ballistic accumulator model (e.g., Annis, Miller, & Palmeri, in press) of perceptual decision making. Recall that these models explain variability in response times and response probabilities via a combination of drift rate, response threshold, starting point, non-decision time (for debates regarding variability in these parameters, see Heathcote, Wagenmakers, & Scott, 2014; Jones & Dzhafarov, 2014; Smith, Ratcliff, & McKoon, 2014).…”