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Appropriate reduction of the posterior distribution in fully Bayesian inversions

Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions and enables us to evaluate both the model parameters and hyperparameters probabilistically by the joint posterior. However, even in a linear inverse problem, it is unsolved how we should extract useful information on the model parameters from the joint posterior. This study presents a theoretical exploration into the appropriate dimensionality reduction of the joint posterior in the fully Bayesian inversion. We classify the ways of probability reduction into the following three categories focused on the marginalisation of the joint posterior: (1) using the joint posterior without marginalisation, (2) using the marginal posterior of the model parameters and (3) using the marginal posterior of the hyperparameters. First, we derive several analytical results that characterise these categories. One is a suite of semianalytic representations of the probability maximisation estimators for respective categories in the linear inverse problem. The mode estimators of categories (1) and (2) are found asymptotically identical for a large number of data and model parameters. We also prove the asymptotic distributions of categories (2) and (3) delta-functionally concentrate on their probability peaks, which predicts two distinct optimal estimates of the model parameters. Second, we conduct a synthetic test and find an appropriate reduction is realised by category (3), typified by Akaike's Bayesian information criterion (ABIC). The other reduction categories are shown inappropriate for the case of many model parameters, where the probability concentration of the marginal posterior of the model parameters is found no longer to mean the central limit theorem...