Interactive statistical decision making (ISDM) features algorithm-dependent data generated through interaction. Existing information-theoretic lower bounds in ISDM largely target expected risk, while tail-sensitive objectives are less developed. We generalize the interactive Fano framework of Chen et al. by replacing the hard success event with a randomized one-bit statistic representing an arbitrary bounded transform of the loss. This yields a Bernoulli f-divergence inequality, which we invert to obtain a two-sided interval for the transform, recovering the previous result as a special case. Instantiating the transform with a bounded hinge and using the Rockafellar-Uryasev representation, we derive lower bounds on the prior-predictive (Bayesian) CVaR of bounded losses. For KL divergence with the mixture reference distribution, the bound becomes explicit in terms of mutual information via Pinsker's inequality.

翻译：交互式统计决策（ISDM）的特点是通过交互生成算法依赖的数据。现有ISDM中的信息论下界主要针对期望风险，而对尾部敏感目标的研究则相对不足。我们推广了Chen等人提出的交互式Fano框架，将硬性成功事件替换为代表损失任意有界变换的随机化单比特统计量。由此导出一个伯努利f-散度不等式，我们通过反演该不等式得到变换量的双侧区间，并将先前结果作为特例恢复。通过采用有界铰链函数作为变换，并利用Rockafellar-Uryasev表示，我们推导出有界损失的先验预测（贝叶斯）CVaR下界。对于以混合参考分布计算KL散度的情形，该下界通过Pinsker不等式可显式表示为互信息的函数。