Generalizing the Fano inequality further

TL;DR

This paper extends the interactive Fano inequality framework to tail-sensitive objectives, providing new lower bounds on Bayesian CVaR of bounded losses using information-theoretic divergences.

Contribution

It generalizes the Fano inequality to arbitrary bounded transforms of loss, enabling tail-sensitive bounds in interactive statistical decision making.

Findings

Derived a Bernoulli f-divergence inequality for tail-sensitive bounds.

Established lower bounds on Bayesian CVaR using the generalized Fano framework.

Connected the bounds to mutual information via Pinsker's inequality.

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.