Krichevsky–Trofimov estimator

In information theory, given an unknown stationary source $\pi$ with alphabet A and a sample w from π, the Krichevsky–Trofimov (KT) estimator produces an estimate pi(w) of the probability of each symbol i &isin; A. This estimator is optimal in the sense that it minimizes the worst-case regret asymptotically.

For a binary alphabet and a string w with m zeroes and n ones, the KT estimator pi(w) is defined as:



\begin{align} p_0(w) &= \frac{m + 1/2}{m + n + 1}, \\[5pt] p_1(w) &= \frac{n + 1/2}{m + n + 1}. \end{align} $$

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