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March 2018 Optimal Gaussian Approximations to the Posterior for Log-Linear Models with Diaconis–Ylvisaker Priors
James Johndrow, Anirban Bhattacharya
Bayesian Anal. 13(1): 201-223 (March 2018). DOI: 10.1214/16-BA1046

Abstract

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.

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James Johndrow. Anirban Bhattacharya. "Optimal Gaussian Approximations to the Posterior for Log-Linear Models with Diaconis–Ylvisaker Priors." Bayesian Anal. 13 (1) 201 - 223, March 2018. https://doi.org/10.1214/16-BA1046

Information

Published: March 2018
First available in Project Euclid: 21 February 2017

zbMATH: 06873724
MathSciNet: MR3737949
Digital Object Identifier: 10.1214/16-BA1046

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Vol.13 • No. 1 • March 2018
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