Approximate Laplace approximations for scalable model selection
arXiv:2012.07429 · doi:10.1111/rssb.12466
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.
References in corpus (11)
- Nearly unbiased variable selection under minimax concave penalty
- Optimal predictive model selection
- A lasso for hierarchical interactions
- Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors
- Bayesian linear regression with sparse priors
- Bayesian variable selection with shrinking and diffusing priors
- Asymptotics for minimisers of convex processes
- Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models
- Uniformly most powerful Bayesian tests
- Scalable Importance Tempering and Bayesian Variable Selection
- Improved Laplace Approximation for Marginal Likelihoods