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

Parameter Collapse due to the Zeros in the Inverse Condition