Computational Bayesian inference using low discrepancy sequences

Abstract

The Integrated Nested Laplace Approximation (INLA) provides fast and accurate Bayesian inference for complex hierarchical models. For INLA, and other deterministic methods, the hyperparameter space is explored and points are laid out in a grid structure. These points are used in some numerical integration scheme for which marginal posterior distributions are computed. The main drawback is that the number of points increase exponentially with the number of hyperparameters. The grid is a type of quasi-Monte Carlo (QMC) point set. Low discrepancy sequences (LDS) are QMC point sets that are well known to have significant advantages over grids in terms of convergence and accuracy, and suffer less from the so-called curse of dimensionality.

This work makes several important contributions. We introduce a new method using LDS to compute marginal posterior distributions for hyperparameters, discuss the convergence properties of the approximations and show that they converge to the true posterior. We also show how these methods can be incorporated into the INLA inference framework, and we outline important extensions that improve the accuracy of our approximations with little extra computational effort needed. Lastly, we build a unique spatio-temporal model of residential crime in Hamilton, using INLA’s stochastic partial differential equation approach to a Log-Gaussian Cox Process, and use an LDS to approximate the latent parameters of the model.

Our results show that for a fixed number of points or computational time, LDS methods can outperform general grid-based methods, leading to better marginal posterior approximations. Modifying the method for the purposes of incorporation to INLA, we show that we can outperform INLA’s grid with respect to computational speed, and obtain accurate and flexible approximations to the model hyperparameters.