Geometric Conditions for the Discrepant Posterior Phenomenon and Connections to Simpson's Paradox
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The discrepant posterior phenomenon (DPP) is a counterintuitive phenomenon that occurs in the Bayesian analysis of multivariate parameters. It refers to when an estimate of a marginal parameter obtained from the posterior is more extreme than both of those obtained using either the prior or the likelihood alone. Inferential claims that exhibit DPP defy intuition, and the phenomenon can be surprisingly ubiquitous in well-behaved Bayesian models. Using point estimation as an example, we derive conditions under which the DPP occurs in Bayesian models with exponential quadratic likelihoods, including Gaussian models and those with local asymptotic normality property, with conjugate multivariate Gaussian priors. We also examine the DPP for the Binomial model, in which the posterior mean is not a linear combination of that of the prior and the likelihood. We provide an intuitive geometric interpretation of the phenomenon and show that there exists a non-trivial 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. Theoretical results are complemented by numerical illustrations. Scenarios covered in this study have implications for parameterization, sensitivity analysis, and prior choice for Bayesian modeling.
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