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The discrepant posterior phenomenon (DPP) is a counter-intuitive phenomenonthat can frequently occur in a Bayesian analysis of multivariate parameters. Itrefers to the phenomenon that a parameter estimate based on a posterior is moreextreme than both of those inferred based on either the prior or the likelihoodalone. Inferential claims that exhibit DPP defy the common intuition that theposterior is a prior-data compromise, and the phenomenon can be surprisinglyubiquitous in well-behaved Bayesian models. In this paper we revisit thisphenomenon and, using point estimation as an example, derive conditions underwhich the DPP occurs in Bayesian models with exponential quadratic likelihoodsand conjugate multivariate Gaussian priors. The family of exponential quadraticlikelihood models includes Gaussian models and those models with localasymptotic normality property. We provide an intuitive geometric interpretationof the phenomenon and show that there exists a nontrivial space of marginaldirections such that the DPP occurs. We further relate the phenomenon to theSimpson's paradox and discover their deep-rooted connection that is associatedwith marginalization. We also draw connections with Bayesian computationalalgorithms when difficult geometry exists. Our discovery demonstrates that DPPis more prevalent than previously understood and anticipated. Theoreticalresults are complemented by numerical illustrations. Scenarios covered in thisstudy have implications for parameterization, sensitivity analysis, and priorchoice for Bayesian modeling.

Geometric Conditions for the Discrepant Posterior Phenomenon and Connections to Simpson's Paradox