Fast, Scalable Approximations to Posterior Distributions in Extended Latent Gaussian Models

TL;DR

This paper introduces a new class of additive models called Extended Latent Gaussian Models, along with a fast, scalable approximate Bayesian inference method that handles complex response distributions and relationships, demonstrated on diverse real-world applications.

Contribution

The paper develops a novel class of models and an efficient inference algorithm that scales to large datasets and complex models, with theoretical error bounds and practical applications.

Findings

The method is faster and scales better than existing approaches.

Approximate posteriors have provably small errors under standard conditions.

Applications demonstrate the method's versatility across different domains.

Abstract

We define a novel class of additive models, called Extended Latent Gaussian Models, that allow for a wide range of response distributions and flexible relationships between the additive predictor and mean response. The new class covers a broad range of interesting models including multi-resolution spatial processes, partial likelihood-based survival models, and multivariate measurement error models. Because computation of the exact posterior distribution is infeasible, we develop a fast, scalable approximate Bayesian inference methodology for this class based on nested Gaussian, Laplace, and adaptive quadrature approximations. We prove that the error in these approximate posteriors is op(1) under standard conditions, and provide numerical evidence suggesting that our method runs faster and scales to larger datasets than methods based on Integrated Nested Laplace Approximations and Markov Chain Monte Carlo, with comparable accuracy. We apply the new method to the mapping of malaria incidence rates in continuous space using aggregated data, mapping leukaemia survival hazards using a Cox Proportional-Hazards model with a continuously-varying spatial process, and estimating the mass of the Milky Way Galaxy using noisy multivariate measurements of the positions and velocities of star clusters in its orbit.