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,π/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 α>d/2, a polynomially enlarged synthetic sample yields ABC contraction at rate n−α/(2α+d) and posterior mean squared risk of order n−2α/(2α+d), matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.
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BibTeX
@misc{nguyen2026minimaxrate,
title = {{On the minimax-rate optimality of approximate Bayesian computation in nonparametric problems}},
author = {Hien Duy Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.15648},
eprint = {2608.15648},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15648},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the minimax-rate optimality of approximate Bayesian computation in nonparametric problems
AU - Hien Duy Nguyen
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.15648
UR - https://arxiv.org/abs/2608.15648
AB - 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,π/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 α>d/2, a polynomially enlarged synthetic sample yields ABC contraction at rate n^−α/(2α+d) and posterior mean squared risk of order n^−2α/(2α+d), matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.
ER -