In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.
Birg ' e @cite_20 gives stronger, but less transparent, bounds than Fano's inequality using Fano-type arguments; again, @math is bounded in terms of an average of Kullback--Leibler divergences, but these are used as the argument for a function, rather than directly substituted. Sason and Verd 'u [Section 3] sasonV18arimoto recently derived a generalized Fano's inequality in terms of the Arimoto-R 'enyi conditional entropy. They also obtained upper bounds on @math in terms of the pairwise R 'enyi divergences [Section 4] sasonV18arimoto .
@cite_20
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Fano's inequality lower bounds the probability of transmission error through a communication channel. Applied to classification problems, it provides a lower bound on the Bayes error rate and motivates the widely used Infomax principle. In modern machine learning, we are often interested in more than just the error rate. In medical diagnosis, different errors incur different cost; hence, the overall risk is cost-sensitive. Two other popular criteria are balanced error rate (BER) and F-score. In this work, we focus on the two-class problem and use a general definition of conditional entropy (including Shannon's as a special case) to derive upper lower bounds on the optimal F-score, BER and cost-sensitive risk, extending Fano's result. As a consequence, we show that Infomax is not suitable for optimizing F-score or cost-sensitive risk, in that it can potentially lead to low F-score and high risk. For cost-sensitive risk, we propose a new conditional entropy formulation which avoids this inconsistency. In addition, we consider the common practice of using a threshold on the posterior probability to tune performance of a classifier. As is widely known, a threshold of 0.5, where the posteriors cross, minimizes error rate--we derive similar optimal thresholds for F-score and BER.