Multinomial and empirical likelihood under convex constraints: Directions of recession, Fenchel duality, the PP algorithm

Abstract

The primal problem of multinomial likelihood maximization restricted to a convex closed subset of the probability simplex is studied. A solution of this problem may assign a positive mass to an outcome with zero count. Such a solution cannot be obtained by the widely used, simplified Lagrange and Fenchel duals. Related flaws in the simplified dual problems, which arise because the recession directions are ignored, are identified and the correct Lagrange and Fenchel duals are developed.

The results permit us to specify linear sets and data such that the empirical likelihood-maximizing distribution exists and is the same as the multinomial likelihood-maximizing distribution. The multinomial likelihood ratio reaches, in general, a different conclusion than the empirical likelihood ratio.

Implications for minimum discrimination information, Lindsay geometry, compositional data analysis, bootstrap with auxiliary information, and Lagrange multiplier test, which explicitly or implicitly ignore information about the support, are discussed.

A solution of the primal problem can be obtained by the PP (perturbed primal) algorithm, that is, as the limit of a sequence of solutions of perturbed primal problems. The PP algorithm may be implemented by the simplified Lagrange or Fenchel dual.