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Mutual information, metric entropy and cumulative relative entropy risk. (English) Zbl 0920.62007

Given a sample of size \(n\) distributed according \(P_{\theta^*}\) let \(\widehat P_t\) be an estimate of \(P_{\theta^*}\) based on the observations up to prime \(t\leq n\). The authors measure the total risk up to time \(n\) by the Kullback-Leibler divergence \(R_{n,\widehat P}(\theta^*)=D_{KL} (P^n_{\theta^*} \|\widehat P)\) where \(\widehat P\) is the product measure of the \(\widehat P_t\). Using this risk the authors introduce the minimax as well as the Bayes strategy. It turns out that the Bayes risk is the mutual information \(I (\Theta^*;Y^n)\) of the random parameter \(\Theta^*\) and the sample \(Y^n\).
Let \(M_{n,\mu}\) be the marginal distribution of \(Y^n\) if \(\Theta^*\) has distribution \(\mu\). The main results in the first part of the paper concern two-sided bounds of \(I(\Theta^*;Y^n)\) and \(D_{KL}(P^n_{\theta^*}\| M_{n,\mu})\) in terms of Renyi distances \(I_\alpha (P_{\theta^*}, P_{\tilde\theta})\) of order \(\alpha\) and \(I_1 (P_{\theta^*},Q^{\tilde\theta})\) where \(Q_\theta\) is the distribution of \(Y\) given \(\theta\). For finite parameter set \(I(\Theta^*;Y^n)\) is shown to tend with exponential rate to the entropy \(H(\Theta^*)\).
The asymptotics of the minimax risk based on the risk \(R_{n,\widehat P}(\theta^*)\) is studied for any \(\Theta\) in the second part of the paper. The authors use the concepts metric entropy and metric dimension to describe the topological structure of \(\Theta\) and to relate these properties to the minimax risk for large \(n\).
Reviewer: F.Liese (Rostock)

MSC:

62B10 Statistical aspects of information-theoretic topics
62C20 Minimax procedures in statistical decision theory
94A15 Information theory (general)
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