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Summary: MEASURES WITH ZEROS IN THE INVERSE OF THEIR
MOMENT MATRIX
J. WILLIAM HELTON, JEAN B. LASSERRE, AND MIHAI PUTINAR
Abstract. We investigate and discuss when the inverse of a multi-
variate truncated moment matrix of a measure µ has zeros in some
prescribed entries. We describe precisely which pattern of these zeroes
corresponds to independence, namely, the measure having a product
structure. A more refined finding is that the key factor forcing a zero
entry in this inverse matrix is a certain conditional triangularity property
of the orthogonal polynomials associated with the measure µ.
1. Introduction
It is well known that zeros in off-diagonal entries of the inverse M-1 of a
n × n covariance matrix M identify pairs of random variables that have no
partial correlation (and so are conditionally independent in case of normally
distributed vectors); see e.g. Wittaker[7, Cor. 6.3.4]. Allowing zeros in the
off-diagonal entries of M-1 is particularly useful for Bayesian estimation of
regression models in statistics, and is called Bayesian covariance selection.
Indeed, estimating a covariance matrix is a difficult problem for large number
of variables, and exploiting sparsity in M-1 may yield efficient methods for
Graphical Gaussian Models (GGM). For more details, the interested reader
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