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Hierarchical Approximate Bayesian Computation

Abstract

Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper, we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function.

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Notes

  1. The Wald distribution describes the behavior of the first-passage time distribution of a single boundary diffusion process.

  2. One can also use a kernel to weigh the fitness of the proposals ξ and \(\theta^{*}_{j}\).

  3. Note that the posterior distributions of θ and ξ will still exist despite a misspecified model. The goal is to estimate the shapes of those posteriors by generating data that is close to the observed data as measured by ρ(X j ,Y j ).

  4. We investigated a range of priors and determined that the choice of priors, if reasonably variable, had little effect on the final estimated posterior. The priors that we selected permit a range of values for d and b that reflect those that are reported in the perceptual and memory literature (Rouder & Lu, 2005; Lee, 2008).

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Correspondence to Brandon M. Turner.

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Turner, B.M., Van Zandt, T. Hierarchical Approximate Bayesian Computation. Psychometrika 79, 185–209 (2014). https://doi.org/10.1007/s11336-013-9381-x

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Key words

  • approximate Bayesian computation
  • hierarchical Bayesian estimation
  • signal detection theory
  • dynamic signal detection