Appropriate reduction of the posterior distribution in fully Bayesian inversions

Summary Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The is also introduced for the hyperparameters in fully inversions enables us to evaluate probabilistically joint posterior. However, even linear inverse problem, it unsolved how we should extract useful on This study presents theoretical exploration into appropriate dimensionality reduction inversion. We classify ways probability following three categories focused marginalization posterior: (1) using without marginalization, (2) marginal (3) First, derive several analytical results that characterize these categories. One suite semi-analytic representations maximization estimators respective problem. mode are found asymptotically identical large number data parameters. prove asymptotic distributions delta-functionally concentrate their peaks, which predicts two distinct optimal estimates Secondly, conduct synthetic test find realized category (3), typified Akaike’s criterion. other shown inappropriate case many parameters, where concentration no longer implies central limit theorem. main cause peaks sharply at underfitted or overfitted solution as increases. exponential growth space model-parameter dimension makes almost-zero-probability events finitely contribute mean be pathological. remedy this pathology counting all realizations integrating over multiplicity. Hence, becomes can conform law numbers with numerous rarity ABIC time complexity ordinary Monte Carlo methods population computations. present geophysical application estimate continuous strain-rate field spatially discrete global navigation satellite system data, demonstrating denser basis function expansions lead oversmoothed naive approaches, while detailed fields resolved convergence (3). often naively believe good constructed finite samples high probabilities, but high-probability domain could inappropriate, exponentially become necessary generating high-dimensional space.