Approximate Bayesian computation using the Fourier integral theorem

Abstract

We present a general method for obtaining posterior samples from a model for which the likelihood function is intractable or otherwise unavailable, but from which simulation is possible. While the inability to evaluate the likelihood impedes the implementation of traditional sampling algorithms, like the Metropolis–Hastings algorithm, our approach, based on the Fourier integral theorem, allows for an exact simulation-based estimate of such likelihoods. Moreover, given its foundations in the Fourier integral theorem, our approach comes with clear guidelines for setting the two tuning parameters on which it relies. Through comparison with alternative methods, specifically synthetic likelihoods and ABC, including both dimension-reduction and full-data approaches, across a variety of examples, culminating in a highly nonlinear state space model based on the Ricker map, we will demonstrate how the Fourier approach is able to provide statistically sensible, full-data likelihood estimation, requiring only that data can be simulated from the model.