标题： 利用全方差定律评估和调整近似推理算法 显示英文标题

标题： Assessment and adjustment of approximate inference algorithms using the law of total variance

摘要： 评估贝叶斯抽样或近似推理方法有效性的常用方法是利用从先验中抽取参数的模拟数据复制。在连续性假设下，对应后验分布中模拟参数值的函数的分位数是均匀分布的。当后验密度被数值近似时检查均匀性可以提供算法有效性的诊断。此外，为实现均匀性所做的调整可以提高近似推理方法的质量。这种方法的一个弱点是似乎难以扩展到感兴趣的标量函数之外。本文开发了一种基于分位数的检查和调整方法的替代方法，该方法本质上是多变量的。新方法基于条件期望的塔性质和总方差定律，用于关联先验和后验期望和协方差。为了进行调整，对近似推理进行了修改，以使正确的先验到后验关系成立。我们在三个示例中展示了该方法。第一个示例在一个无需似然的推理问题中使用了辅助模型。第二个示例考虑了深度神经网络广义线性混合模型中的变分贝叶斯近似修正。我们的最终应用考虑了一个深度神经网络代理，用于近似高斯过程回归预测推断。 显示英文摘要

摘要： A common method for assessing validity of Bayesian sampling or approximate inference methods makes use of simulated data replicates for parameters drawn from the prior. Under continuity assumptions, quantiles of functions of the simulated parameter values within corresponding posterior distributions are uniformly distributed. Checking for uniformity when a posterior density is approximated numerically provides a diagnostic for algorithm validity. Furthermore, adjustments to achieve uniformity can improve the quality of approximate inference methods. A weakness of this general approach is that it seems difficult to extend beyond scalar functions of interest. The present article develops an alternative to quantile-based checking and adjustment methods which is inherently multivariate. The new approach is based on use of the tower property of conditional expectation and the law of total variance for relating prior and posterior expectations and covariances. For adjustment, approximate inferences are modified so that the correct prior to posterior relationships hold. We illustrate the method in three examples. The first uses an auxiliary model in a likelihood-free inference problem. The second considers corrections for variational Bayes approximations in a deep neural network generalized linear mixed model. Our final application considers a deep neural network surrogate for approximating Gaussian process regression predictive inference.