An Extended Laplace Approximation Method for Bayesian Inference of Self-Exciting Spatial-Temporal Models of Count Data

TL;DR

This paper introduces an extended Laplace approximation method with higher-order corrections for Bayesian inference in self-exciting spatio-temporal count models, reducing bias in parameter estimation for large datasets.

Contribution

It develops a sixth-order correction to the Laplace approximation for more accurate Bayesian inference in complex self-exciting spatial-temporal models.

Findings

Extended LA reduces bias in parameter estimates.

Performance demonstrated on simulated and real data.

Limited bias remains in small parameter spaces.

Abstract

Self-Exciting models are statistical models of count data where the probability of an event occurring is influenced by the history of the process. In particular, self-exciting spatio-temporal models allow for spatial dependence as well as temporal self-excitation. For large spatial or temporal regions, however, the model leads to an intractable likelihood. An increasingly common method for dealing with large spatio-temporal models is by using Laplace approximations (LA). This method is convenient as it can easily be applied and is quickly implemented. However, as we will demonstrate in this manuscript, when applied to self-exciting Poisson spatial-temporal models, Laplace Approximations result in a significant bias in estimating some parameters. Due to this bias, we propose using up to sixth-order corrections to the LA for fitting these models. We will demonstrate how to do this in a Bayesian setting for Self-Exciting Spatio-Temporal models. We will further show there is a limited parameter space where the extended LA method still has bias. In these uncommon instances we will demonstrate how a more computationally intensive fully Bayesian approach using the Stan software program is possible in those rare instances. The performance of the extended LA method is illustrated with both simulation and real-world data.