Compatibility-based Bayesian inference: asymptotic limits, variational principles, and Wasserstein geometry

Abstract

Approximate Bayesian Computation (ABC) replaces intractable likelihood evaluations with comparisons between observed and simulated data. We develop a measure-theoretic formulation of discrepancy-based Bayesian inference in which compatibility is encoded through nonnegative weights. For vanishing thresholds, we establish convergence of the corresponding posterior distributions in total variation. In the large-sample regime, we identify conditions under which the posterior converges to the prior conditioned on an asymptotic compatibility region, showing that concentration need not occur at a fixed positive tolerance. We also derive persistence bounds under weaker assumptions and prove stability of the associated entropy-regularized variational problems. For Wasserstein discrepancies, we obtain a geometric representation through distances from compatibility balls and an exact variational reduction to probability distributions over model laws.

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