The is no other model or hypothesis verification tool in Bayesian statistics that is as widely used as the Bayes factor. We focus on generative models that are likelihood-free and, therefore, render the computation of Bayes factors (marginal likelihood ratios) far from obvious. We propose a deep learning estimator of the Bayes factor based on simulated data from two competing models using the likelihood ratio trick. This estimator is devoid of summary statistics and obviates some of the difficulties with ABC model choice. We establish sufficient conditions for consistency of our Deep Bayes Factor estimator as well as its consistency as a model selection tool. We investigate the performance of our estimator on various examples using a wide range of quality metrics related to estimation and model decision accuracy. After training, our deep learning approach enables rapid evaluations of the Bayes factor estimator at any fictional data arriving from either hypothesized model, not just the observed data $Y_0$. This allows us to inspect entire Bayes factor distributions under the two models and to quantify the relative location of the Bayes factor evaluated at $Y_0$ in light of these distributions. Such tail area evaluations are not possible for Bayes factor estimators tailored to $Y_0$. We find the performance of our Deep Bayes Factors competitive with existing MCMC techniques that require the knowledge of the likelihood function. We also consider variants for posterior or intrinsic Bayes factors estimation. We demonstrate the usefulness of our approach on a relatively high-dimensional real data example about determining cognitive biases.

翻译：在贝叶斯统计学中，没有任何其他模型或假设验证工具能像贝叶斯因子那样被广泛使用。我们专注于无似然生成模型，这使得贝叶斯因子（边际似然比）的计算远非显而易见。我们提出了一种基于模拟数据的深度学习贝叶斯因子估计器，该估计器利用似然比技巧，从两个竞争模型中生成数据。该估计器无需摘要统计量，并避免了近似贝叶斯计算模型选择中的一些困难。我们为深度贝叶斯因子估计器的一致性及其作为模型选择工具的一致性建立了充分条件。我们使用与估计和模型决策准确性相关的多种质量指标，在各种示例上研究了该估计器的性能。训练完成后，我们的深度学习方法能够快速评估来自任一假设模型（而不仅仅是观测数据$Y_0$）的任何虚构数据的贝叶斯因子估计值。这使我们能够检查两个模型下的完整贝叶斯因子分布，并根据这些分布量化在$Y_0$处评估的贝叶斯因子的相对位置。这种尾部区域评估对于针对$Y_0$定制的贝叶斯因子估计器是不可能的。我们发现，深度贝叶斯因子的性能与需要已知似然函数的现有马尔可夫链蒙特卡洛技术相当。我们还考虑了用于后验或内在贝叶斯因子估计的变体。我们通过一个关于确定认知偏见的相对高维真实数据示例，展示了我们方法的实用性。