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.
翻译:我们研究了贝叶斯框架中的数据驱动决策问题,贝叶斯框架中的预期值被风险敏感昆虫风险计量办法所取代。我们侧重于计算后继物分布十分棘手的问题,这是使用大型数据集和复杂数据生成模型的现代应用中典型的典型情况。我们运用了一种新型的对风险敏感的变异贝叶斯框架(RSVB),以共同计算一种对风险敏感的远地点近似和相应的决定规则。拟议的RSVB框架可用于提取计算方法,以进行风险敏感近似贝叶斯人的推断。我们表明,我们的总框架包括两种众所周知的计算方法,即近似巴伊西亚的误判,即天性VB和损失分类的VB。我们还研究了这些计算近似于风险的远地点和相应的决定规则对预测性表现的影响。我们比较了RSVB近似近似远地点的近似近似远地点以及相应的最佳价值和决定规则的趋同率。我们用三个例子来说明我们在准和无parasian模型和多度的模型环境中的理论模型结论性结论性结论性和多种模型。我们用三个例子来说明。