A post by Jack Simons, PhD student on the Compass programme.

# Introduction

I began my PhD with my supervisors, Dr Song Liu and Professor Mark Beaumont with the intention of combining their respective fields of research; Density Ratio Estimation (DRE), and Simulation Based Inference (SBI):

- DRE is a rapidly growing paradigm in machine learning which (broadly) provides efficient methods of comparing densities without the need to compute each density individually. For a comprehensive yet accessible overview of DRE in Machine Learning see [1].
- SBI is a group of methods which seek to solve Bayesian inference problems when the likelihood function is intractable. If you wish for a concise overview of the current work, as well as motivation then I recommend [2].

Last year we released a paper, * Variational Likelihood-Free Gradient Descent * [3] which combined these fields. This blog post seeks to condense, and make more accessible, the contents of the paper.

# Motivation: Likelihood-Free Inference

Let’s begin by introducing likelihood-free inference. We wish to do inference on the posterior distribution of parameters for a specific observation , i.e. we wish to infer which can be decomposed via Bayes’ rule as

The likelihood-free setting is that, additional to the usual intractability of the normalising constant in the denominator, the likelihood, , is also intractable. In lieu of this, we require an implicit likelihood which describes the relation between data and parameters in the form of a forward model/simulator (hence *simulation* based inference!). (more…)