# Adaptive Approximate Bayesian Computation Tolerance Selection

## TL;DR

This paper introduces an adaptive method for selecting tolerances in ABC-PMC algorithms, enhancing efficiency and automating stopping criteria for better posterior sampling in complex models.

## Contribution

It proposes an automatic, adaptive tolerance selection and stopping rule for ABC-PMC, improving computational efficiency and ease of implementation.

## Key findings

- Adaptive tolerance selection improves sampling efficiency.
- Automated stopping rule facilitates termination of sampling.
- Examples demonstrate significant efficiency gains.

## Abstract

Approximate Bayesian Computation (ABC) methods are increasingly used for inference in situations in which the likelihood function is either computationally costly or intractable to evaluate. Extensions of the basic ABC rejection algorithm have improved the computational efficiency of the procedure and broadened its applicability. The ABC-Population Monte Carlo (ABC-PMC) approach of Beaumont et al. (2009) has become a popular choice for approximate sampling from the posterior. ABC-PMC is a sequential sampler with an iteratively decreasing value of the tolerance, which specifies how close the simulated data need to be to the real data for acceptance. We propose a method for adaptively selecting a sequence of tolerances that improves the computational efficiency of the algorithm over other common techniques. In addition we define a stopping rule as a by-product of the adaptation procedure, which assists in automating termination of sampling. The proposed automatic ABC-PMC algorithm can be easily implemented and we present several examples demonstrating its benefits in terms of computational efficiency.