  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.
