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:^[1]

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

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

See also

• Rule of succession
• Bayesian inference using conjugate priors for the categorical distribution

References

1. 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.