Generalized Iterative Scaling for Log-Linear Models

Say that a probability distribution \pᵢ; i I\ over a finite set I is in "product form" if (1) pᵢ = ᵢ ᵈₒ=₁ ₛ^bₛi where ᵢ and \bₒ₈\ are given constants and where and \ₛ\ are determined from the equations (2) ₈ ₈ bₒ₈ pᵢ = kₛ, s = 1, 2, , d; (3) ₈ ₈ pᵢ = 1. Probability distributions in product form arise from minimizing the discriminatory information ₈ ₈ pᵢ pᵢ/ᵢ subject to (2) and (3) or from maximizing entropy or maximizing likelihood. The theory of the iterative scaling method of determining (1) subject to (2) and (3) has, until now, been limited to the case when bₒ₈ = 0, 1. In this paper the method is generalized to allow the bₒ₈ to be any real numbers. This expands considerably the list of probability distributions in product form which it is possible to estimate by maximum likelihood.