On the Statistical Consistency of Risk-Sensitive Bayesian Decision-Making

TL;DR

This paper introduces a novel risk-sensitive variational Bayesian framework that addresses intractable posterior computations in Bayesian decision-making, providing theoretical convergence guarantees and practical algorithms for complex models.

Contribution

The paper develops a new RSVB framework that unifies existing methods and analyzes its convergence and impact on decision quality in complex Bayesian models.

Findings

RSVB includes naive VB and loss-calibrated VB as special cases

Convergence rates of the RSVB posterior and decision rules are established

Framework demonstrated in newsvendor and Gaussian process classification examples

Abstract

We study data-driven decision-making problems in the Bayesian framework, where the expectation in the Bayes risk is replaced by a risk-sensitive entropic risk measure. We focus on problems where calculating the posterior distribution is intractable, a typical situation in modern applications with large datasets and complex data generating models. We leverage a dual representation of the entropic risk measure to introduce a novel risk-sensitive variational Bayesian (RSVB) framework for jointly computing a risk-sensitive posterior approximation and the corresponding decision rule. The proposed RSVB framework can be used to extract computational methods for doing risk-sensitive approximate Bayesian inference. We show that our general framework includes two well-known computational methods for doing approximate Bayesian inference viz. naive VB and loss-calibrated VB. We also study the impact of these computational approximations on the predictive performance of the inferred decision rules and values. We compute the convergence rates of the RSVB approximate posterior and also of the corresponding optimal value and decision rules. We illustrate our theoretical findings in both parametric and nonparametric settings with the help of three examples: the single and multi-product newsvendor model and Gaussian process classification.