Duality of I Projections and Maximum Likelihood Estimation for Log-Linear Models under Cone Constraints

摘要

Important order restrictions and general log-linear models for multi-nominal experiments can often be expressed by requiring that the vector composed of the logs of the probabilities fall within a closed convex cone. Here is exhibited a duality relationship (by way of a Fenchel duality theorem) between these types of problems and projections in I-divergence geometry. Many cone-constrained maximum likelihood estimation problems are exactly equivalent to an I projection onto a translation of the negative polar cone. This duality relationship permits the concise characterization of cone-restricted maximum likelihood estimates and the use of the iterative algorithms which are analogous to the iterative proportional-fitting procedure. A Fenchel duality theorem to exhibit an elegant duality relationship between cone-constrained, multi-nominal maximum likelihood problems and I projections of the Kullback-Liebler type. This duality relationship is often useful for obtaining maximum likelihood estimates (MLE's) of the probability vector p of a multi-nominal distribution under restrictions that require the vector 1n p= (1n p1,...,1n pk) to lie within a closed convex cone in k-dimensional Euclidian space. Reprints. (jhd)