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Both resources in the natural environment and concepts in a semantic spaceare distributed "patchily", with large gaps in between the patches. To describepeople's internal and external foraging behavior, various random walk modelshave been proposed. In particular, internal foraging has been modeled assampling: in order to gather relevant information for making a decision, peopledraw samples from a mental representation using random-walk algorithms such asMarkov chain Monte Carlo (MCMC). However, two common empirical observationsargue against simple sampling algorithms such as MCMC. First, the spatialstructure is often best described by a L\'evy flight distribution: theprobability of the distance between two successive locations follows apower-law on the distances. Second, the temporal structure of the sampling thathumans and other animals produce have long-range, slowly decaying serialcorrelations characterized as $1/f$-like fluctuations. We propose that mentalsampling is not done by simple MCMC, but is instead adapted to multimodalrepresentations and is implemented by Metropolis-coupled Markov chain MonteCarlo (MC$^3$), one of the first algorithms developed for sampling frommultimodal distributions. MC$^3$ involves running multiple Markov chains inparallel but with target distributions of different temperatures, and it swapsthe states of the chains whenever a better location is found. Heated chainsmore readily traverse valleys in the probability landscape to propose moves tofar-away peaks, while the colder chains make the local steps that explore thecurrent peak or patch. We show that MC$^3$ generates distances betweensuccessive samples that follow a L\'evy flight distribution and $1/f$-likeserial correlations, providing a single mechanistic account of these twopuzzling empirical phenomena.

Mental Sampling in Multimodal Representations