  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, 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.
