期刊名称：Discussion Papers / School of Business, University of New South Wales
出版年度：2007
卷号：2007
出版社：Sydney
摘要：Estimating a covariance matrix efficiently and discovering its structure are important
statistical problems with applications in many fields. This article takes a Bayesian
approach to estimate the covariance matrix of Gaussian data. We use ideas from Gaussian
graphical models and model selection to construct a prior for the covariance matrix that is a
mixture over all decomposable graphs, where a graph means the configuration of nonzero offdiagonal
elements in the inverse of the covariance matrix. Our prior for the covariance matrix
is such that the probability of each graph size is specified by the user and graphs of equal size
are assigned equal probability. Most previous approaches assume that all graphs are equally
probable. We give empirical results that show the prior that assigns equal probability over
graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying
the correct decomposable graph and in more efficiently estimating the covariance matrix.
The advantage is greatest when the number of observations is small relative to the dimension
of the covariance matrix. The article also shows empirically that there is minimal change in statistical
efficiency in using the mixture over decomposable graphs prior for estimating a general
covariance compared to the Bayesian estimator by Wong et al. (2003), even when the graph of
the covariance matrix is nondecomposable. However, our approach has some important advantages
over that of Wong et al. (2003). Our method requires the number of decomposable
graphs for each graph size. We show how to estimate these numbers using simulation and that
the simulation results agree with analytic results when such results are known. We also show
how to estimate the posterior distribution of the covariance matrix using Markov chain Monte
Carlo with the elements of the covariance matrix integrated out and give empirical results that
show the sampler is computationally efficient and converges rapidly. Finally, we note that both
the prior and the simulation method to evaluate the prior apply generally to any decomposable
graphical model.
