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

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 $o_p(1)$ 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 leukemia 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. Supplementary materials for this article are available online.

KEYWORDS:

• Additive model
• Approximate inference
• Bayesian methods
• Laplace approximation
• Spatial statistics

Supplementary Materials

The proof of Theorem 1 is given in the Online Supplementary Materials.

Acknowledgments

The authors are grateful for helpful comments provided by two referees and an associate editor, as well as Blair Bilodeau, Yanbo Tang, and Gwen Eadie. Parts of this manuscript appeared in the first author’s Ph.D. thesis for which he is grateful for comments from Gwen Eadie, Linbo Wang, Nancy Reid, and Anthony Davison. This work was supported by the Natural Sciences and Engineering Research Council of Canada’s Discovery Grant and Postgraduate Scholarships programs.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors gratefully acknowledge funding from the Natural Sciences and Engineering Research Council of Canada.