Generalizing the Fano inequality further
Abstract
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.
Cite
@article{arxiv.2601.12027,
title = {Generalizing the Fano inequality further},
author = {Raghav Bongole and Tobias J. Oechtering and Mikael Skoglund},
journal= {arXiv preprint arXiv:2601.12027},
year = {2026}
}