Parameter Collapse due to the Zeros in the Inverse Condition

TL;DR

This paper explores how zeros in the inverse condition of the truncated moment matrix lead to parameter collapse into product structures or conditional independence in probability measures, supported by algorithms for computation.

Contribution

It provides examples and algorithms demonstrating how zeros in the inverse condition induce parameter collapse to product measures or conditional independence.

Findings

Zeros in the inverse condition imply measure factorization.

Parameter restrictions lead to simplified, product-structured models.

Algorithms for computing zeros in the inverse condition are developed.

Abstract

Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/0702314) expose the relationship between three properties of a measure: the conditional triangularity property of the associated orthogonal polynomials, the zeros in the inverse condition of the truncated moment matrix, and conditional independence. The purpose of this article is to provide examples of parameter collapse to product structure given that the zeros in the inverse condition holds up to some degree d. Specifically, start with a parameterized family of probability density functions; require that the zeros in the inverse condition up to degree d holds; and validate that imposing this restriction on the parameterized family results in a measure with product structure, or at least that conditional independence holds. Algorithms related to parameter collapse are supplied, including the computation of the zeros in the inverse condition up to degree d.