An extended empirical saddlepoint approximation for intractable likelihoods
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 being 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, available on CRAN.
What the paper shows and why it matters (AI-generated)
Synthetic Likelihood makes inference tractable for models whose likelihood can't be written down, but it leans on an assumption — that summary statistics are roughly normally distributed — that doesn't always hold. This paper relaxes that assumption with a more flexible density estimator, tested on a complex individual-based forest model where few other inference methods work at all, and packaged as the esaddle R package on CRAN. It's a narrow methods niche, but a persistent one: the approach keeps resurfacing in later work on Bayesian synthetic likelihood and empirical-likelihood approaches to approximate Bayesian computation.