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.
Using Fano's inequality, Yang and Barron @cite_2 obtained order-optimal minimax risk lower bounds that depend only on global metric entropy features of the underlying function class, without explicitly constructing a packing set. The required metric entropy features (bounds on the packing number and covering number) are available from results in approximation theory for many function classes of interest. Guntuboyina @cite_1 obtained a lower bound on the average error probability in terms of general @math -divergences, and also generalized the metric entropy results of Yang and Barron @cite_2 to certain @math -divergences such as the @math -divergence. An interesting direction for future work would be to obtain a non-asymptotic result analogous to Theorem for the case where only the global metric entropy features are available.
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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. In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard l p norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n) H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F 0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space time approximation tradeoff curve.Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model. Lower bounds involving f-divergences between the underlying probability measures are proved for the minimax risk in estimation problems. Our proofs just use simple convexity facts. Special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as Fano's inequality, Pinsker's inequality and inequalities based on global entropy conditions. Two applications are provided: a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix. We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size @math is unknown and may be comparable with or even much larger than the number of observations @math . We treat the respective regions where the functional is nonsmooth and smooth separately. In the nonsmooth regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the smooth regime, we apply a bias-corrected version of the maximum likelihood estimator (MLE). We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: 1) the entropy @math and 2) @math . We obtain the minimax @math rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity @math for entropy estimation. We also demonstrate that the sample complexity for estimating @math n S^ 1 S @math 1 , we show the minimax @math rate for estimating @math is @math for infinite support size, while the maximum @math rate for the MLE is @math . For all the above cases, the behavior of the minimax rate-optimal estimators with @math samples is essentially that of the MLE (plug-in rule) with @math samples, which we term “effective sample size enlargement.” We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow–Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable.