Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework

TL;DR

This paper introduces a novel automatic proposal distribution for ABC-MCMC algorithms based on quasi-likelihood modeling of summary statistics, improving efficiency in Bayesian inference with intractable likelihoods.

Contribution

It develops a quasi-likelihood based approach to construct proposal distributions for ABC-MCMC, extending QL theory to vector parameters and variable variance scenarios.

Findings

Effective proposal distribution for ABC-MCMC demonstrated

Improved sampling efficiency in examples and real data

Extension of QL theory to multivariate parameters

Abstract

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy between prior and posterior. Therefore, more elaborate methods, such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be used. However, the elaboration of a proposal density for MCMC is a sensitive issue and very difficult in the ABC setting, where the likelihood is intractable. We discuss an automatic proposal distribution useful for ABC-MCMC algorithms. This proposal is inspired by the theory of quasi-likelihood (QL) functions and is obtained by modelling the distribution of the summary statistics as a function of the parameters. Essentially, given a real-valued vector of summary statistics, we reparametrize the model by means of a regression function of the statistics on parameters, obtained by sampling from the original model in a pilot-run simulation study. The QL theory is well established for a scalar parameter, and it is shown that when the conditional variance of the summary statistic is assumed constant, the QL has a closed-form normal density. This idea of constructing proposal distributions is extended to non constant variance and to real-valued parameter vectors. The method is illustrated by several examples and by an application to a real problem in population genetics.