arXiv:2105.01962v4 Announce Type: replace Abstract: Approximate Bayesian Computation (ABC) replaces intractable likelihood evaluations with comparisons between observed and simulated data. We formulate discrepancy-based inference through nonnegative compatibility weights. Prior-integrated convergence of rescaled weights gives total variation convergence as the tolerance vanishes. For a fixed positive tolerance and increasing sample size, we give a general representation-based formulation of the conditional-prior limit known from ABC and coarsened Bayesian inference, and establish convergence in expected total variation under stabilization in probability and a prior-null boundary condition. We also derive acceptance-persistence bounds under partial stabilization. The Gibbs variational identity yields convergence of minimum values and stability of approximate minimizers; the fixed-tolerance limit is characterized separately by constrained relative-entropy minimization. For additional penalty-based functionals of the limiting Wasserstein discrepancy, we prove an exact reduction to distributions over model laws and, under a geodesic assumption, a representation by distances from a Wasserstein ball. These penalty-based updates are distinguished from the original fixed-tolerance ABC limit and illustrated through a Gaussian comparison.