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Generalized variational inference (GVI) provides an optimization-theoreticframework for statistical estimation that encapsulates many traditionalestimation procedures. The typical GVI problem is to compute a distribution ofparameters that maximizes the expected payoff minus the divergence of thedistribution from a specified prior. In this way, GVI enables likelihood-freeestimation with the ability to control the influence of the prior by tuning theso-called learning rate. Recently, GVI was shown to outperform traditionalBayesian inference when the model and prior distribution are misspecified. Inthis paper, we introduce and analyze a new GVI formulation based on utilitytheory and risk management. Our formulation is to maximize the expected payoffwhile enforcing constraints on the maximizing distribution. We recover theoriginal GVI distribution by choosing the feasible set to include a constrainton the divergence of the distribution from the prior. In doing so, weautomatically determine the learning rate as the Lagrange multiplier for theconstraint. In this setting, we are able to transform the infinite-dimensionalestimation problem into a two-dimensional convex program. This reformulationfurther provides an analytic expression for the optimal density of parameters.In addition, we prove asymptotic consistency results for empiricalapproximations of our optimal distributions. Throughout, we draw connectionsbetween our estimation procedure and risk management. In fact, we demonstratethat our estimation procedure is equivalent to evaluating a risk measure. Wetest our procedure on an estimation problem with a misspecified model and priordistribution, and conclude with some extensions of our approach.

A Risk Management Perspective on Statistical Estimation and Generalized Variational Inference