Minimax filtering regret via relations between information and estimation

Albert No, Tsachy Weissman

Abstract: We investigate the problem of continuous-time causal estimation under a minimax criterion. Let X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> = {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> , 0 ≤ t ≤ T} be governed by probability law P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sub> from some class of possible laws indexed by θ ∈ S, and Y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> be the noise corrupted observations of X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> available to the estimator. We characterize the estimator minimizing the worst case regret, where regret is the difference between the expected loss of the estimator and that optimized for the true law of X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> . We then relate this 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 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. Also, the optimum minimax estimator is the Bayesian estimator assuming the capacity-achieving prior. Moreover, we show that this minimax estimator is not only minimizing the worst case regret but also essentially minimizing the regret for “most” of the other sources in the uncertainty set. We present a couple of examples for the construction of an approximately minimax filter via an approximation of the associated capacity achieving distribution.