The discrepant posterior phenomenon (DPP) is a counter-intuitive phenomenon that can frequently occur in a Bayesian analysis of multivariate parameters. It refers to the phenomenon that a parameter estimate based on a posterior is more extreme than both of those inferred based on either the prior or the likelihood alone. Inferential claims that exhibit DPP defy the common intuition that the posterior is a prior-data compromise, and the phenomenon can be surprisingly ubiquitous in well-behaved Bayesian models. In this paper we revisit this phenomenon and, using point estimation as an example, derive conditions under which the DPP occurs in Bayesian models with exponential quadratic likelihoods and conjugate multivariate Gaussian priors. The family of exponential quadratic likelihood models includes Gaussian models and those models with local asymptotic normality property. We provide an intuitive geometric interpretation of the phenomenon and show that there exists a nontrivial space of marginal directions such that the DPP occurs. We further relate the phenomenon to the Simpson's paradox and discover their deep-rooted connection that is associated with marginalization. We also draw connections with Bayesian computational algorithms when difficult geometry exists. Our discovery demonstrates that DPP is more prevalent than previously understood and anticipated. Theoretical results are complemented by numerical illustrations. Scenarios covered in this study have implications for parameterization, sensitivity analysis, and prior choice for Bayesian modeling.
翻译:相异的事后现象(DPP)是一种反直觉现象,在对多种变异参数进行的巴伊西亚分析中经常出现。它指的是一种现象,即基于后一种参数的参数估计比根据先前或单独可能性推论的两种推论都更加极端。推论中,显示DPP的指称无视共同直觉,即后一种是先前数据妥协,而这一现象在熟知的巴伊西亚模型中可能是令人惊讶的无处不在的。在本文中,我们重新审视了这一现象,并以点估计为例,得出了在巴伊西亚模型中以指数二次变异可能性和共变多变多变前两种模型进行计算的条件。指数四变可能性模型的组别包括Gausian模型和具有本地无症状正常特性的模型。我们提供了对这一现象的直观几何解释,并表明DPP的边际方向存在非边际空间。我们进一步将这一现象与Simbiscalal化现象联系起来,而我们之前的常态变化和深层次的变法分析也比我们所理解的地基变法分析更深。