Title: Mental Sampling in Multimodal Representations,
Abstract: Both resources in the natural environment and concepts in a semantic space
are distributed "patchily", with large gaps in between the patches. To describe
people's internal and external foraging behavior, various random walk models
have been proposed. In particular, internal foraging has been modeled as
sampling: in order to gather relevant information for making a decision, people
draw samples from a mental representation using random-walk algorithms such as
Markov chain Monte Carlo (MCMC). However, two common empirical observations
argue against simple sampling algorithms such as MCMC. First, the spatial
structure is often best described by a Lévy flight distribution: the
probability of the distance between two successive locations follows a
power-law on the distances. Second, the temporal structure of the sampling that
humans and other animals produce have long-range, slowly decaying serial
correlations characterized as $1/f$-like fluctuations. We propose that mental
sampling is not done by simple MCMC, but is instead adapted to multimodal
representations and is implemented by Metropolis-coupled Markov chain Monte
Carlo (MC$^3$), one of the first algorithms developed for sampling from
multimodal distributions. MC$^3$ involves running multiple Markov chains in
parallel but with target distributions of different temperatures, and it swaps
the states of the chains whenever a better location is found. Heated chains
more readily traverse valleys in the probability landscape to propose moves to
far-away peaks, while the colder chains make the local steps that explore the
current peak or patch. We show that MC$^3$ generates distances between
successive samples that follow a Lévy flight distribution and $1/f$-like
serial correlations, providing a single mechanistic account of these two
puzzling empirical phenomena.