An Extended Empirical Saddlepoint Approximation for Intractable Likelihoods

TL;DR

This paper introduces a flexible density estimator for Synthetic Likelihood that better captures non-normal data distributions, improving inference accuracy in complex stochastic models across ecology, biology, and genetics.

Contribution

The paper proposes the Extended Empirical Saddlepoint approximation as a novel, scalable density estimator for Synthetic Likelihood, relaxing the normality assumption and enhancing inference accuracy.

Findings

The new estimator captures large departures from normality.

It is scalable to high-dimensional data.

It improves parameter estimation accuracy.

Abstract

The challenges posed by complex stochastic models used in computational ecology, biology and genetics have stimulated the development of approximate approaches to statistical inference. Here we focus on Synthetic Likelihood (SL), a procedure that reduces the observed and simulated data to a set of summary statistics, and quantifies the discrepancy between them through a synthetic likelihood function. SL requires little tuning, but it relies on the approximate normality of the summary statistics. We relax this assumption by proposing a novel, more flexible, density estimator: the Extended Empirical Saddlepoint approximation. In addition to proving the consistency of SL, under either the new or the Gaussian density estimator, we illustrate the method using two examples. One of these is a complex individual-based forest model for which SL offers one of the few practical possibilities for statistical inference. The examples show that the new density estimator is able to capture large departures from normality, while being scalable to high dimensions, and this in turn leads to more accurate parameter estimates, relative to the Gaussian alternative. The new density estimator is implemented by the esaddle R package, which can be found on the Comprehensive R Archive Network (CRAN).