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%20has%20emerged%20as%20a%20popular%20method%20for%20approximate%20inference%20for%20high-dimensional%20Bayesian%20models.%20In%20this%20paper%2C%20we%20propose%20a%20novel%20VI%20method%20that%20extends%20the%20naive%20mean%20field%20via%20entropic%20regularization%2C%20referred%20to%20as%20%24Xi%24-variational%20inference%20(%24Xi%24-VI).%20%24Xi%24-VI%20has%20a%20close%20connection%20to%20the%20entropic%20optimal%20transport%20problem%20and%20benefits%20from%20the%20computationally%20efficient%20Sinkhorn%20algorithm.%20We%20show%20that%20%24Xi%24-variational%20posteriors%20effectively%20recover%20the%20true%20posterior%20dependency%2C%20where%20the%20dependence%20is%20downweighted%20by%20the%20regularization%20parameter.%20We%20analyze%20the%20role%20of%20dimensionality%20of%20the%20parameter%20space%20on%20the%20accuracy%20of%20%24Xi%24-variational%20approximation%20and%20how%20it%20affects%20computational%20considerations%2C%20providing%20a%20rough%20characterization%20of%20the%20statistical-computational%20trade-off%20in%20%24Xi%24-VI.%20We%20also%20investigate%20the%20frequentist%20properties%20of%20%24Xi%24-VI%20and%20establish%20results%20on%20consistency%2C%20asymptotic%20normality%2C%20high-dimensional%20asymptotics%2C%20and%20algorithmic%20stability.%20We%20provide%20sufficient%20criteria%20for%20achieving%20polynomial-time%20approximate%20inference%20using%20the%20method.%20Finally%2C%20we%20demonstrate%20the%20practical%20advantage%20of%20%24Xi%24-VI%20over%20mean-field%20variational%20inference%20on%20simulated%20and%20real%20data.)
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