Approximate Laplace approximations for scalable model selection

TL;DR

This paper introduces the approximate Laplace approximation (ALA), a computationally efficient method for Bayesian model selection that maintains strong theoretical properties and is applicable to various complex models.

Contribution

The paper presents ALA, a novel approximation method that reduces computational cost in Bayesian model selection while preserving consistency and applicability to diverse models.

Findings

ALA reduces computation to solving least-squares problems.

ALA achieves model selection consistency in generalized models.

The method is effective in high-dimensional and misspecified models.

Abstract

We propose the approximate Laplace approximation (ALA) to evaluate integrated likelihoods, a bottleneck in Bayesian model selection. The Laplace approximation (LA) is a popular tool that speeds up such computation and equips strong model selection properties. However, when the sample size is large or one considers many models the cost of the required optimizations becomes impractical. ALA reduces the cost to that of solving a least-squares problem for each model. Further, it enables efficient computation across models such as sharing pre-computed sufficient statistics and certain operations in matrix decompositions. We prove that in generalized (possibly non-linear) models ALA achieves a strong form of model selection consistency for a suitably-defined optimal model, at the same functional rates as exact computation. We consider fixed- and high-dimensional problems, group and hierarchical constraints, and the possibility that all models are misspecified. We also obtain ALA rates for Gaussian regression under non-local priors, an important example where the LA can be costly and does not consistently estimate the integrated likelihood. Our examples include non-linear regression, logistic, Poisson and survival models. We implement the methodology in the R package mombf.