Risk-Sensitive Variational Bayes: Formulations and Bounds

We study data-driven decision-making problems in a parametrized Bayesian framework. We adopt a risk-sensitive approach to modeling the interplay between statistical estimation of parameters and optimization, by computing a risk measure over a loss/disutility function with respect to the posterior distribution over the parameters. While this forms the standard Bayesian decision-theoretic approach, we focus on problems where calculating the posterior distribution is intractable, a typical situation in modern applications with %high-dimensional parameter space large datasets, heterogeneity due to observed covariates and latent group structure. The key methodological innovation we introduce in this paper is to leverage a dual representation of the risk measure to introduce an optimization-based framework for approximately computing the posterior risk-sensitive objective, as opposed to using standard sampling based methods such as Markov Chain Monte Carlo. Our analytical contributions include rigorously proving finite sample bounds on the `optimality gap' of optimizers obtained using the computational methods in this paper, from the `true' optimizers of a given decision-making problem. We illustrate our results by comparing the theoretical bounds with simulations of a newsvendor problem on two methods extracted from our computational framework.