  Variational inference (VI) has emerged as a popular method for approximate
inference for high-dimensional Bayesian models. In this paper, we propose a
novel VI method that extends the naive mean field via entropic regularization,
referred to as $\Xi$-variational inference ($\Xi$-VI). $\Xi$-VI has a close
connection to the entropic optimal transport problem and benefits from the
computationally efficient Sinkhorn algorithm. We show that $\Xi$-variational
posteriors effectively recover the true posterior dependency, where the
dependence is downweighted by the regularization parameter. We analyze the role
of dimensionality of the parameter space on the accuracy of $\Xi$-variational
approximation and how it affects computational considerations, providing a
rough characterization of the statistical-computational trade-off in $\Xi$-VI.
We also investigate the frequentist properties of $\Xi$-VI and establish
results on consistency, asymptotic normality, high-dimensional asymptotics, and
algorithmic stability. We provide sufficient criteria for achieving
polynomial-time approximate inference using the method. Finally, we demonstrate
the practical advantage of $\Xi$-VI over mean-field variational inference on
simulated and real data.
