Thermodynamic integration via differential evolution: A method for estimating marginal likelihoods

A typical goal in cognitive psychology is to select the model that provides the best explanation of the observed behavioral data. The Bayes factor provides a principled approach for making these selections, though the integral required to calculate the marginal likelihood for each model is intractable for most cognitive models. In these cases, Monte Carlo techniques must be used to approximate the marginal likelihood, such as thermodynamic integration (TI; Friel & Pettitt, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(3), 589-607 2008; Lartillot & Philippe, Systematic Biology, 55(2), 195-207 2006), which relies on sampling from the posterior at different powers (called power posteriors). TI can become computationally expensive when using population Markov chain Monte Carlo (MCMC) approaches such as differential evolution MCMC (DE-MCMC; Turner et al., Psychological Methods, 18(3), 368 2013) that require several interacting chains per power posterior. Here, we propose a method called thermodynamic integration via differential evolution (TIDE), which aims to reduce the computational burden associated with TI by using a single chain per power posterior (R code available at https://osf.io/ntmgw/ ). We show that when applied to non-hierarchical models, TIDE produces an approximation of the marginal likelihood that closely matches TI. When extended to hierarchical models, we find that certain assumptions about the dependence between the individual- and group-level parameters samples (i.e., dependent/independent) have sizable effects on the TI approximated marginal likelihood. We propose two possible extensions of TIDE to hierarchical models, which closely match the marginal likelihoods obtained through TI with dependent/independent sampling in many, but not all, situations. Based on these findings, we believe that TIDE provides a promising method for estimating marginal likelihoods, though future research should focus on a detailed comparison between the methods of estimating marginal likelihoods for cognitive models.