# Bayesian Indirect Inference Using a Parametric Auxiliary Model

arXiv: https://arxiv.org/abs/1505.03372 · HTML: https://mlsanity.com/paper/1505.03372/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2015.

## Built on by (most cited first, 7 of 7)

- [Generative Modeling: A Review](https://mlsanity.com/paper/2501.05458/) (2024)
- [Wasserstein Gaussianization and Efficient Variational Bayes for Robust Bayesian Synthetic Likelihood](https://mlsanity.com/paper/2305.14746/) (2023)
- [Bayesian Semi-supervised Inference via a Debiased Modeling Approach](https://mlsanity.com/paper/2509.17385/) (2025)
- [Bayesian Semiparametric Causal Inference: Targeted Doubly Robust Estimation of Treatment Effects](https://mlsanity.com/paper/2511.15904/) (2025)
- [Validation and Inference of Agent Based Models](https://mlsanity.com/paper/2107.03619/) (2021)
- [Pseudo-Likelihood Inference](https://mlsanity.com/paper/2311.16656/) (2023)
- [Simulation-Based Inference via Regression Projection and Batched Discrepancies](https://mlsanity.com/paper/2602.03613/) (2026)

## Abstract (arXiv)

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.