  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 in relatively
small samples. The proposed approximation provides a computationally scalable
and principled approach to regularized estimation and approximate Bayesian
inference for log-linear models.
