The University of Sydney Business School, Sydney, Australia.
School of Mathematics and Statistics, University of Wollongong, Wollongong, Australia.
Behav Res Methods. 2021 Jun;53(3):1148-1165. doi: 10.3758/s13428-020-01348-w.
Recent advances in Markov chain Monte Carlo (MCMC) extend the scope of Bayesian inference to models for which the likelihood function is intractable. Although these developments allow us to estimate model parameters, other basic problems such as estimating the marginal likelihood, a fundamental tool in Bayesian model selection, remain challenging. This is an important scientific limitation because testing psychological hypotheses with hierarchical models has proven difficult with current model selection methods. We propose an efficient method for estimating the marginal likelihood for models where the likelihood is intractable, but can be estimated unbiasedly. It is based on first running a sampling method such as MCMC to obtain samples for the model parameters, and then using these samples to construct the proposal density in an importance sampling (IS) framework with an unbiased estimate of the likelihood. Our method has several attractive properties: it generates an unbiased estimate of the marginal likelihood, it is robust to the quality and target of the sampling method used to form the IS proposals, and it is computationally cheap to estimate the variance of the marginal likelihood estimator. We also obtain the convergence properties of the method and provide guidelines on maximizing computational efficiency. The method is illustrated in two challenging cases involving hierarchical models: identifying the form of individual differences in an applied choice scenario, and evaluating the best parameterization of a cognitive model in a speeded decision making context. Freely available code to implement the methods is provided. Extensions to posterior moment estimation and parallelization are also discussed.
近年来，马尔可夫链蒙特卡罗（MCMC）方法的进展扩展了贝叶斯推断的范围，使其适用于似然函数难以计算的模型。尽管这些发展允许我们估计模型参数，但其他基本问题，如估计边际似然，这是贝叶斯模型选择的基本工具，仍然具有挑战性。这是一个重要的科学限制，因为使用分层模型检验心理假设，使用当前的模型选择方法证明是困难的。我们提出了一种有效的方法来估计难以计算但可以无偏估计的似然函数的边际似然。它基于首先运行 MCMC 等采样方法来获得模型参数的样本，然后使用这些样本在重要性抽样（IS）框架中构建提案密度，其中包含似然的无偏估计。我们的方法具有几个吸引人的特性：它生成边际似然的无偏估计，对用于形成 IS 提案的采样方法的质量和目标具有鲁棒性，并且估计边际似然估计量的方差的计算成本低廉。我们还获得了该方法的收敛特性，并提供了最大化计算效率的指南。该方法在两个涉及分层模型的具有挑战性的案例中得到了说明：识别应用选择场景中个体差异的形式，以及在快速决策背景下评估认知模型的最佳参数化。提供了可免费实现这些方法的代码。还讨论了对后验矩估计和并行化的扩展。
