  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.
