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

原文摘要

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 of the correspondence. First, maximum likelihood estimates for parameters in a monotonic score model proposed by Agresti, Chuang, and Kezouh (1987) are characterized using the duality relationships established in this article. Second, we use the correspondence established in this article to identify a rather surprising link between two different types of positive dependency. The iterative proportional fitting procedure is also explained in the context of the aforementioned duality relationship as a special case of an iterative I-projection algorithm given by Csiszar (1975).

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