Krichevsky–Trofimov estimator
# Krichevsky–Trofimov estimator

In information theory, given an unknown stationary source π with alphabet A and a sample w from π, the Krichevsky–Trofimov (KT) estimator produces an estimate $p_i(w)$ of the probability of each symbol i ∈ 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 $p_i(w)$ is defined as:

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

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

