Student Perspectives: Application of Density Ratio Estimation to Likelihood-Free problems

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 $\theta$ for a specific observation $x=x_{\mathrm{obs}}$, i.e. we wish to infer $p(\theta|x_{\mathrm{obs}})$ which can be decomposed via Bayes’ rule as

$p(\theta|x_{\mathrm{obs}}) = \frac{p(x_{\mathrm{obs}}|\theta)p(\theta)}{\int p(x_{\mathrm{obs}}|\theta)p(\theta) \mathrm{d}\theta}.$

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