# On the use of approximate Bayesian computation Markov chain Monte Carlo with inflated tolerance and post-correction

## TL;DR

This paper introduces an adaptive ABC-MCMC method using inflated tolerance and post-correction to improve inference accuracy and efficiency for complex models with intractable likelihoods.

## Contribution

It proposes a novel adaptive ABC-MCMC algorithm that automatically balances tolerance levels and incorporates post-processing for better estimators and confidence intervals.

## Key findings

- Post-processing estimators outperform direct targeting at fine tolerances.
- The proposed confidence intervals are reliable.
- The adaptive algorithm achieves accurate inference with minimal user input.

## Abstract

Approximate Bayesian computation allows for inference of complicated probabilistic models with intractable likelihoods using model simulations. The Markov chain Monte Carlo implementation of approximate Bayesian computation is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We consider an approach using a relatively large tolerance for the Markov chain Monte Carlo sampler to ensure its sufficient mixing, and post-processing the output leading to estimators for a range of finer tolerances. We introduce an approximate confidence interval for the related post-corrected estimators, and propose an adaptive approximate Bayesian computation Markov chain Monte Carlo, which finds a `balanced' tolerance level automatically, based on acceptance rate optimisation. Our experiments show that post-processing based estimators can perform better than direct Markov chain targetting a fine tolerance, that our confidence intervals are reliable, and that our adaptive algorithm leads to reliable inference with little user specification.