The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model—a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

中文翻译：

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桥接采样教程

边际似然在贝叶斯统计的许多领域中起着重要作用，例如参数估计、模型比较和模型平均。然而，在大多数应用中，边际似然在分析上是不易于处理的，必须使用数值方法来近似。在这里，我们提供了有关桥接采样的教程（Bennett，1976 年；Meng & Wong，1996 年），这是一种可靠且相对简单的采样方法，允许研究人员获得不同复杂度模型的边际可能性。首先，我们使用 beta-二项式模型作为运行示例介绍桥采样和三种相关的采样方法。然后，我们应用桥采样来估计期望价 (EV) 模型（一种流行的强化学习模型）的边际可能性。我们的结果表明，桥采样为单个参与者和 EV 模型的分层版本提供了准确的估计。我们得出结论，桥采样对于数学心理学家来说是一种有吸引力的方法，他们通常旨在近似一组有限的可能高维模型的边际可能性。