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We discuss probabilistic models of random covariance structures defined bydistributions over sparse eigenmatrices. The decomposition of orthogonalmatrices in terms of Givens rotations defines a natural, interpretableframework for defining distributions on sparsity structure of randomeigenmatrices. We explore theoretical aspects and implications for conditionalindependence structures arising in multivariate Gaussian models, and discussconnections with sparse PCA, factor analysis and Gaussian graphical models.Methodology includes model-based exploratory data analysis and Bayesiananalysis via reversible jump Markov chain Monte Carlo. A simulation studyexamines the ability to identify sparse multivariate structures compared to thebenchmark graphical modelling approach. Extensions to multivariate normalmixture models with additional measurement errors move into the framework oflatent structure analysis of broad practical interest. We explore theimplications and utility of the new models with summaries of a detailed appliedstudy of a 20-dimensional breast cancer genomics data set.

Models of random sparse eigenmatrices matrices and Bayesian analysis of multivariate structure