Partially factorized variational inference for high-dimensional mixed models
针对高维广义线性混合模型，标准均值场变分推断会严重低估后验不确定性，本文提出部分因子化方法，在保证不确定性量化精度的同时，计算成本与参数和观测数呈线性关系，并提供了理论保证和R包实现。
Summary While generalized linear mixed models are a fundamental tool in applied statistics, many specifications, such as those involving categorical factors with many levels or interaction terms, can be computationally challenging to estimate due to the need to compute or approximate high-dimensional integrals. Variational inference is a popular way to perform such computations, especially in the Bayesian context. However, naive use of such methods can provide unreliable uncertainty quantification. We show that this is indeed the case for mixed models, proving that standard mean-field variational inference dramatically underestimates posterior uncertainty in high dimensions. We then show how appropriately relaxing the mean-field assumption leads to methods whose uncertainty quantification does not deteriorate in high dimensions, and whose total computational cost scales linearly with the number of parameters and observations. Our theoretical and numerical results focus on mixed models with Gaussian or binomial likelihoods, and rely on connections to random graph theory to obtain sharp high-dimensional asymptotic analysis. We also provide generic results, which are of independent interest, relating the accuracy of variational inference to the convergence rate of the corresponding coordinate ascent algorithm that is used to find it. Our proposed methodology is implemented in the R package vglmer. Numerical results with simulated and real data examples illustrate the favourable computation cost versus accuracy trade-off of our approach compared to various alternatives.