Stochastic interpretation for the Arimoto algorithm

The Arimoto algorithm computes the Gallager function max_Q E₀(ρ, Q) for a given channel P (y | x) and parameter ρ, by means of alternating maximization. Along the way, it generates a sequence of input distributions Q₁(x), Q₂(x), ..., that converges to the maximizing input Q*(x). We propose a stochastic interpretation for the Arimoto algorithm. We show that for a random (i.i.d.) codebook with a distribution Q_k(x), the next distribution Q_k+1(x) in the Arimoto algorithm is equal to the type (Q´) of the feasible transmitted codeword that maximizes the conditional Gallager exponent (conditioned on a specific transmitted codeword type Q´). This interpretation is a first step toward finding a stochastic mechanism for on-line channel input adaptation.