Bayesian Indirect Inference Using a Parametric Auxiliary Model

TL;DR

This paper compares parametric Bayesian indirect inference methods, specifically ABC II and pBIL, providing theoretical insights, assumptions, and practical applications for complex models with intractable likelihoods.

Contribution

It introduces a unified Bayesian indirect likelihood framework and offers new theoretical results distinguishing ABC II from pBIL, with practical examples illustrating their use.

Findings

pBIL is fundamentally different from ABC II.

Theoretical insights into pBIL's behavior and assumptions.

Applications to complex quantile distributions and macroparasite models.

Abstract

Indirect inference (II) is a methodology for estimating the parameters of an intractable (generative) model on the basis of an alternative parametric (auxiliary) model that is both analytically and computationally easier to deal with. Such an approach has been well explored in the classical literature but has received substantially less attention in the Bayesian paradigm. The purpose of this paper is to compare and contrast a collection of what we call parametric Bayesian indirect inference (pBII) methods. One class of pBII methods uses approximate Bayesian computation (referred to here as ABC II) where the summary statistic is formed on the basis of the auxiliary model, using ideas from II. Another approach proposed in the literature, referred to here as parametric Bayesian indirect likelihood (pBIL), uses the auxiliary likelihood as a replacement to the intractable likelihood. We show that pBIL is a fundamentally different approach to ABC II. We devise new theoretical results for pBIL to give extra insights into its behaviour and also its differences with ABC II. Furthermore, we examine in more detail the assumptions required to use each pBII method. The results, insights and comparisons developed in this paper are illustrated on simple examples and two other substantive applications. The first of the substantive examples involves performing inference for complex quantile distributions based on simulated data while the second is for estimating the parameters of a trivariate stochastic process describing the evolution of macroparasites within a host based on real data. We create a novel framework called Bayesian indirect likelihood (BIL) that encompasses pBII as well as general ABC methods so that the connections between the methods can be established.