Approximate Bayesian computation (ABC) gives exact results under the assumption of model error

TL;DR

This paper demonstrates that ABC algorithms can produce exact Bayesian inference results when assuming a uniform additive model error, clarifying their interpretation and guiding their application.

Contribution

It shows that ABC algorithms yield exact results under a uniform model error assumption and introduces a generalization replacing the fixed cutoff with a variable acceptance probability.

Findings

ABC algorithms give exact results with sufficient summaries under model error assumptions.

The acceptance probability can be adapted to incorporate general error distributions.

ABC can be viewed as calibration for implicit stochastic models.

Abstract

Approximate Bayesian computation (ABC) or likelihood-free inference algorithms are used to find approximations to posterior distributions without making explicit use of the likelihood function, depending instead on simulation of sample data sets from the model. In this paper we show that under the assumption of the existence of a uniform additive model error term, ABC algorithms give exact results when sufficient summaries are used. This interpretation allows the approximation made in many previous application papers to be understood, and should guide the choice of metric and tolerance in future work. ABC algorithms can be generalized by replacing the 0-1 cut-off with an acceptance probability that varies with the distance of the simulated data from the observed data. The acceptance density gives the distribution of the error term, enabling the uniform error usually used to be replaced by a general distribution. This generalization can also be applied to approximate Markov chain Monte Carlo algorithms. In light of this work, ABC algorithms can be seen as calibration techniques for implicit stochastic models, inferring parameter values in light of the computer model, data, prior beliefs about the parameter values, and any measurement or model errors.