文章基本信息

标题：Optimal Gaussian Approximations to the Posterior for Log-Linear Models with Diaconis–Ylvisaker Priors
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作者：James Johndrow ; Anirban Bhattacharya
期刊名称：Bayesian Analysis
印刷版ISSN：1931-6690
电子版ISSN：1936-0975
出版年度：2018
卷号：13
期号：1
页码：201-223
DOI：10.1214/16-BA1046
语种：English
出版社：International Society for Bayesian Analysis
摘要：In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis–Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. Here we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis–Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback–Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even for modest sample sizes. We also propose a method for model selection using the approximation. The proposed approximation provides a computationally scalable approach to regularized estimation and approximate Bayesian inference for log-linear models.
关键词：credible region; conjugate prior; contingency table;Dirichet–Multinomial; Kullback–Leibler divergence; Laplace approximation.