# Multilevel rejection sampling for approximate Bayesian computation

## TL;DR

This paper introduces a multilevel rejection sampling method for approximate Bayesian computation that enhances computational efficiency while maintaining the accuracy of traditional rejection sampling.

## Contribution

It proposes a novel multilevel Monte Carlo approach to accelerate likelihood-free Bayesian inference via rejection sampling.

## Key findings

- Significant reduction in computational cost compared to standard rejection sampling.
- Maintains unbiased, independent samples from the approximate posterior.
- Effective variance reduction demonstrated through experiments.

## Abstract

Likelihood-free methods, such as approximate Bayesian computation, are powerful tools for practical inference problems with intractable likelihood functions. Markov chain Monte Carlo and sequential Monte Carlo variants of approximate Bayesian computation can be effective techniques for sampling posterior distributions in an approximate Bayesian computation setting. However, without careful consideration of convergence criteria and selection of proposal kernels, such methods can lead to very biased inference or computationally inefficient sampling. In contrast, rejection sampling for approximate Bayesian computation, despite being computationally intensive, results in independent, identically distributed samples from the approximated posterior. An alternative method is proposed for the acceleration of likelihood-free Bayesian inference that applies multilevel Monte Carlo variance reduction techniques directly to rejection sampling. The resulting method retains the accuracy advantages of rejection sampling while significantly improving the computational efficiency.