Iterative proportional scaling via decomposable submodels for contingency tables

Abstract

We propose iterative proportional scaling (IPS) via decomposable submodels for maximizing likelihood function of a hierarchical model for contingency tables. In ordinary IPS the proportional scaling is performed by cycling through the members of the generating class of a hierarchical model. We propose to adjust more marginals at each step. This is accomplished by expressing the generating class as a union of decomposable submodels and cycling through the decomposable models. We prove convergence of our proposed procedure, if the amount of scaling is adjusted properly at each step. We also analyze the proposed algorithms around the maximum likelihood estimate (MLE) in detail. Faster convergence of our proposed procedure is illustrated by numerical examples.

Rational partition models under iterative proportional scaling

“…The IPS algorithm is widely used in survey statistics, such as in [9,27,43], and has been researched in connection to optimal transport problems in its more restricted form as Sinkhorn's algorithm [6,37]. Partition models include hierarchical models, which have been the subject of much study in connection with the IPS algorithm [21,25,42]. Maximum likelihood estimation for log-linear models and its relationship to algebraic and combinatorial objects is also of interest in algebraic statistics.…”