A novel method of marginalisation using low discrepancy sequences for integrated nested Laplace approximations

Recently, it has been shown that the shape of a marginal distribution can be more accurately and efficiently captured using a set of low discrepancy sequence (LDS) points compared to standard grid points. This suggests that the use of LDS could improve the approximation to marginal posterior distributions produced by grid-based Bayesian methods such as the Integrated Nested Laplace Approximation (INLA). However, obtaining marginal posteriors using LDS is not straightforward. Two algorithms are proposed that can be incorporated into the INLA implementation to approximate marginal posterior distributions using LDS without sacrificing computational efficiency. Two examples are also presented to demonstrate that the proposed algorithms, when used inside INLA, can estimate marginal posteriors more accurately and efficiently than the grid approximation INLA employs. A distinct advantage is that these algorithms can also capture multimodal shapes that the current numerical integration free algorithm (NIFA) method used by INLA cannot.