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Krichevsky–Trofimov estimator

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Ininformation theory, given an unknownstationary sourceπ with alphabetA and a samplew fromπ, theKrichevsky–Trofimov (KT) estimator produces an estimatep_i(w) of the probability of each symboli ∈ A. This estimator is optimal in the sense that it minimizes the worst-caseregret asymptotically.

For a binary alphabet and a stringw withm zeroes andn ones, the KT estimatorp_i(w) is defined as:^[1]

{\begin{aligned}p_{0}(w)&={\frac {m+1/2}{m+n+1}},\\[5pt]p_{1}(w)&={\frac {n+1/2}{m+n+1}}.\end{aligned}}

This corresponds to the posterior mean of aBeta-Bernoulli posterior distribution with prior $1/2$ .For the general case the estimate is made using aDirichlet-Categorical distribution.

References

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^Krichevsky, R. E.; Trofimov, V. K. (1981). "The Performance of Universal Encoding".IEEE Trans. Inf. Theory. IT-27 (2):199–207.doi:10.1109/TIT.1981.1056331.

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