Would You Cross a River that is Only 4 Foot Deep on Average? A simulation study of worst-case Bayesian approximability.

Abstract

In cognitive science, many cognitive functions are currently being modelled using Bayesian (or probabilistic) models. These models often give good descriptive fit to performance data obtained in the lab. Yet, the Bayesian modelling framework faces a theoretical problem: many Bayesian computations are known to be NPhard, and thus it is unclear how the computations postulated by Bayesian models can be tractable for resource-bounded minds/brains like our own. A common proposal by Bayesian modellers in cognitive science has been to suggest that the problem of intractability can be overcome by assuming that human minds/brains use approximation algorithms to approximate (NP-hard) Bayesian computations. In this paper we investigate this proposal using computer simulations of a particular approximation algorithm (Gibbs sampling) for a particular Bayesian model (Blokpoel, Kwisthout, van der Weide, & van Rooij, 2010) as a case study. We will show that, even though approximation may look like a solution at first glance, further investigation proves this assumption wrong.