On the minimax-rate optimality of approximate Bayesian computation in nonparametric problems

Abstract

Approximate Bayesian computation (ABC) replaces likelihood evaluation by simulation and comparison of observed and synthetic data. We establish minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates. The contraction theorem uses local prior mass, bounds on ABC acceptance probabilities, and control of prior mass outside a sieve. In fixed-design orthogonal-series regression with centered $g$-and-$k$ errors, an infinite Gaussian series prior with a compact scale hyperprior yields minimax-rate contraction and a minimax-rate clipped posterior mean. In compound Poisson decompounding, only random sums are observed and the target is the underlying jump density. With an unknown count intensity in a fixed compact subinterval of $(0,\pi/2)$, we prove stability of the zero-count-augmented trigonometric population summaries and use a square-root Gaussian series prior on the space of probability density functions. Over bounded periodic Sobolev classes of smoothness $\alpha>d/2$, a polynomially enlarged synthetic sample yields ABC contraction at rate $n^{-\alpha/(2\alpha+d)}$ and posterior mean squared risk of order $n^{-2\alpha/(2\alpha+d)}$, matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.