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.

翻译：我们建议使用大致拉普尔近似(ALA)来评估综合可能性、巴伊西亚模式选择中的瓶颈。拉普尔近近似(LA)是一个常用工具,可以加速这种计算,并装备强大的模型选择属性。然而,当样本规模大或认为许多模型时,所需的优化成本就变得不切实际。拉普尔近近近近近近(ALA)可以将成本降低到解决每种模型的最低方问题的成本。此外,它还能够有效地计算出各种模型,例如共享预先计算过的充分统计数据和矩阵分解中的某些操作。我们证明,在通用(可能非线性)模型中,ALA实现了一种强烈的模型选择一致性,其功能率与精确计算相同。我们考虑到固定和高方的问题、组和等级限制,以及所有模型被错误描述的可能性。我们还获得了非本地前期高方回归率的ALA率,这是一个重要的例子,使得LA成本可能很高,并且不连贯地估计综合的可能性。我们的例子包括非线性回归、物流、Poisson和生存模型。我们在MOMb中采用的方法。