On asymptotic validity of naive inference with an approximate likelihood

TL;DR

This paper establishes conditions under which naive inference using an approximate likelihood retains the same first-order asymptotic properties as inference with the true likelihood, with applications to Laplace and reduced dependence approximations.

Contribution

It provides theoretical conditions ensuring asymptotic validity of inference with approximate likelihoods, extending understanding of when such methods are reliable.

Findings

Conditions for asymptotic validity of naive inference with approximate likelihoods

Application to Laplace approximation in latent variable models

Application to reduced dependence approximations in Ising models

Abstract

Many statistical models have likelihoods which are intractable: it is impossible or too expensive to compute the likelihood exactly. In such settings, a common approach is to replace the likelihood with an approximation, and proceed with inference as if the approximate likelihood were the exact likelihood. In this paper, we describe conditions on the approximate likelihood which guarantee that this naive inference with an approximate likelihood has the same first-order asymptotic properties as inference with the exact likelihood. We investigate the implications of these results for inference using a Laplace approximation to the likelihood in a simple two-level latent variable model, and using reduced dependence approximations to the likelihood in an Ising model on a lattice.