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Minimax Filtering Regret via Relations Between Information and Estimation

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2 Author(s)
Albert No ; Dept. of Electr. Eng., Stanford Univ., Stanford, CA, USA ; Tsachy Weissman

We investigate the problem of continuous-time causal estimation under a minimax criterion. Let XT = (Xt,0 ≤ t ≤ T) be governed by the probability law Pθ from a class of possible laws indexed by θ ∈ A, and YT be the noise corrupted observations of XT 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.

Published in:

IEEE Transactions on Information Theory  (Volume:60 ,  Issue: 8 )