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In high-dimensions, the prior tails can have a significant effect on bothposterior computation and asymptotic concentration rates. To achieve optimalrates while keeping the posterior computations relatively simple, an empiricalBayes approach has recently been proposed, featuring thin-tailed conjugatepriors with data-driven centers. While conjugate priors ease some of thecomputational burden, Markov chain Monte Carlo methods are still needed, whichcan be expensive when dimension is high. In this paper, we develop avariational approximation to the empirical Bayes posterior that is fast tocompute and retains the optimal concentration rate properties of the original.In simulations, our method is shown to have superior performance compared toexisting variational approximations in the literature across a wide range ofhigh-dimensional settings.

Variational approximations of empirical Bayes posteriors in high-dimensional linear models