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

Abstract Important order restrictions and general log-linear models for multinomial experiments can often be expressed by requiring that the vector composed of the logs of the probabilities fall within a closed convex cone. Here we exhibit a duality relationship (by way of a Fenchel duality theorem) between these types of problems and projections in I-divergence geometry. We show that 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 iterative algorithms proposed by Csiszar (1975) and Dykstra (1985b), which are analogous to the iterative proportional-fitting procedure. Computational aspects of maximum likelihood estimates are discussed, and a relationship with least squares projections onto isotonic cones is presented. We discuss examples in detail to exhibit the usefulness and power...