A hypergraph-theoretic analysis of collapsibility and decomposability for extended log-linear models

  • Authors:

  • F. M. Malvestuto

  • Affiliations:

  • Dip. Di Scienze Dell'Informazione, Università “La Sapienza” Di Roma, Via Salaria 113, I-00198, Roma, Italy. mal@dsi.uniromal.it

  • Venue:
  • Statistics and Computing
  • Year:
  • 2001

Quantified Score

Hi-index 0.00

Visualization

Abstract

Extended log-linear models (ELMs) are the natural generalization of log-linear models when the positivity assumption is relaxed. The hypergraph language, which is currently used to specify the syntax of ELMs, both provides an insight into key notions of the theory of ELMs such as collapsibility and decomposability, and allows to work out efficient algorithms to solve some problems of inference. This is the case for the three search problems addressed in this paper and referred to as the approximation problem, the selective-reduction problem and the synthesis problem. The approximation problem consists in finding the smallest decomposable ELM that contains a given ELM and is such that the given ELM is collapsible onto each of its generators. The selective-reduction problem consists in deleting the maximum number of generators of a given ELM in such a way that the resulting ELM is a submodel and none of certain variables of interest is missing. The synthesis problem consists in finding a minimal ELM containing the intersection of ELMs specified by given independence relations. We show that each of the three search problems above can be reduced to an equivalent search problem on hypergraphs, which can be solved in polynomial time.