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 π_i(w) of the probabilities 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(m, n) can be defined recursively:

${\begin{aligned}P(0,0)&=1,\\P(m,n+1)&=P(m,n){\dfrac {n+1/2}{m+n+1}},\\P(m+1,n)&=P(m,n){\dfrac {m+1/2}{m+n+1}}.\end{aligned}}$

See also

• Rule of succession
• Dirichlet-multinomial distribution