NDLI: Minimax Filtering Regret via Relations Between Information and Estimation

Content Provider	IEEE Xplore Digital Library
Author	No, A. Weissman, T.
Copyright Year	1963
Abstract	We investigate the problem of continuous-time causal estimation under a minimax criterion. Let X^T = (X_t,0 ≤ t ≤ T) be governed by the probability law P_θ from a class of possible laws indexed by θ ∈ A, and Y^T be the noise corrupted observations of X^T available to the estimator. We characterize the estimator minimizing the worst case regret, where regret is the difference between the causal estimation loss of the estimator and that of the optimum estimator. One of the main contributions of this paper is characterizing the minimax estimator, showing that it is in fact a Bayesian estimator. We then relate minimax regret to the channel capacity when the channel is either Gaussian or Poisson. In this case, we characterize the minimax regret and the minimax estimator more explicitly. If we further assume that the uncertainty set consists of deterministic signals, the worst case regret is exactly equal to the corresponding channel capacity, namely the maximal mutual information attainable across the channel among all possible distributions on the uncertainty set of signals. The corresponding minimax estimator is the Bayesian estimator assuming the capacity-achieving prior. Using this relation, we also show that the capacity achieving prior coincides with the least favorable input. In addition, we show that this minimax estimator is not only minimizing the worst case regret, but also essentially minimizing regret for most of the other sources in the uncertainty set. We present a couple of examples for the construction of a minimax filter via an approximation of the associated capacity achieving distribution.
Sponsorship	IEEE Information Theory Society
Starting Page	4832
Ending Page	4847
Page Count	16
File Size	524756
File Format	PDF
ISSN	00189448
Volume Number	60
Issue Number	8
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-01-01
Publisher Place	U.S.A.
Access Restriction	One Nation One Subscription (ONOS)
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Bayes methods Uncertainty Mutual information Channel capacity Estimation error Noise least favorable input Mismatched estimation minimax regret regret-capacity strong regret-capacity directed information sparse signal estimation AWGN channel poisson channel
Content Type	Text
Resource Type	Article
Subject	Library and Information Sciences Information Systems Computer Science Applications

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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