arXiv:2610.28153v1 Announce Type: new Abstract: We develop an information-theoretic method for lower bounding the prior-predictive conditional value-at-risk (CVaR) of loss in interactive statistical decision making (ISDM). A generalized Fano inequality replaces a single error event by a randomized binary statistic constructed from a bounded transform of the loss, reducing the problem to a Bernoulli $f$-divergence inequality. A bounded hinge transform yields a direct lower bound on Bayesian CVaR under a prior fixed in advance. We derive reverse-KL specializations and obtain explicit bounds for Gaussian mean estimation and Gaussian stochastic bandits. We then impose local differential privacy (LDP) at the observation stage of the same ISDM model. LDP contraction inequalities combined with the adaptive KL chain rule give explicit Bayesian CVaR lower bounds for Gaussian mean estimation with heterogeneous privacy parameters and Gaussian $K$-armed bandits.