Department of Psychology, University of Amsterdam, Amsterdam, Netherlands.
Department of Psychology, Vanderbilt University, Nashville, TN, 37235, USA.
Behav Res Methods. 2019 Apr;51(2):930-947. doi: 10.3758/s13428-018-1172-y.
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.
认知心理学中的一个典型目标是选择能够对观察到的行为数据提供最佳解释的模型。贝叶斯因子为进行这些选择提供了一种原则性的方法，尽管对于大多数认知模型来说，计算每个模型边际似然所需的积分是难以处理的。在这些情况下，必须使用蒙特卡罗技术来近似边际似然，例如热力学积分（TI；Friel & Pettitt，《皇家统计学会杂志：B 辑（统计方法）》，70（3），589-607，2008；Lartillot & Philippe，《系统生物学》，55（2），195-207，2006），它依赖于在不同幂次（称为幂后验）下从后验中进行采样。当使用需要为每个幂后验交互多个链的群体马尔可夫链蒙特卡罗（MCMC）方法（如差分进化 MCMC（DE-MCMC；Turner 等人，《心理方法》，18（3），368，2013））时，TI 可能会变得计算昂贵。在这里，我们提出了一种称为通过差分进化的热力学积分（TIDE）的方法，该方法旨在通过为每个幂后验使用单个链来减少与 TI 相关的计算负担（可在 https://osf.io/ntmgw/ 获得 R 代码）。我们表明，当应用于非层次模型时，TIDE 生成的边际似然近似值与 TI 非常匹配。当扩展到层次模型时，我们发现个体和群体水平参数样本之间的依赖关系（即，依赖/独立）的某些假设对 TI 近似边际似然有很大影响。我们提出了两种将 TIDE 扩展到层次模型的方法，它们在许多情况下（但不是所有情况）都与通过 TI 获得的边际似然非常匹配，而与依赖/独立采样有关。基于这些发现，我们相信 TIDE 为估计边际似然提供了一种很有前途的方法，尽管未来的研究应该集中在对认知模型的边际似然估计方法的详细比较上。
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