Bayesian Analysis

Simple Marginally Noninformative Prior Distributions for Covariance Matrices

Alan Huang and M. P. Wand

Full-text: Open access


A family of prior distributions for covariance matrices is studied. Members of the family possess the attractive property of all standard deviation and correlation parameters being marginally noninformative for particular hyperparameter choices. Moreover, the family is quite simple and, for approximate Bayesian inference techniques such as Markov chain Monte Carlo and mean field variational Bayes, has tractability on par with the Inverse-Wishart conjugate family of prior distributions. A simulation study shows that the new prior distributions can lead to more accurate sparse covariance matrix estimation.

Article information

Bayesian Anal., Volume 8, Number 2 (2013), 439-452.

First available in Project Euclid: 24 May 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian inference Gibbs sampling Markov chain Monte Carlo Mean field variational Bayes


Huang, Alan; Wand, M. P. Simple Marginally Noninformative Prior Distributions for Covariance Matrices. Bayesian Anal. 8 (2013), no. 2, 439--452. doi:10.1214/13-BA815.

Export citation


  • Armagan, A., Dunson, D. B., and Clyde, M. (2011). “Generalized Beta Mixtures of Gaussians.” In Shawe-Taylor, J., Zemel, R., Bartlett, P., Pereira, F., and Weinberger, K. (eds.), Advances in Neural Information Processing Systems 24, 523–531.
  • Barnard, J., McCulloch, R., and Meng, X. L. (2000). “Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage.” Statistica Sinica, 10: 1281–1311.
  • Bien, J. and Tibshirani, R. J. (2011). “Sparse estimation of a covariance matrix.” Biometrika, 98: 807–820.
  • Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Reading, MA: Addison-Wesley.
  • Diggle, P., Heagerty, P., Liang, K.-L., and Zeger, S. (2002). Analysis of Longitudinal Data. New York: Cambridge University Press, 2 edition.
  • Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models.” Bayesian Analysis, 1: 515–533.
  • Mathai, A. M. (2005). “A pathway to matrix-variate gamma and normal densities.” Linear Algebra and Its Applications, 396: 317–328.
  • Menictas, M. and Wand, M. (2013). “Variational inference for marginal longitudinal semiparametric regression.” Stat, 2: 61–71.
  • O’Malley, A. and Zaslavsky, A. (2008). “Domain-level covariance analysis for multilevel survey data with structured nonresponse.” Journal of the American Statistical Association, 103: 1405–1418.
  • Spiegelhalter, D. J., Thomas, A., Best, N. G., Gilks, W. R., and Lunn, D. (2003). BUGS: Bayesian inference using Gibbs sampling. Medical Research Council Biostatistics Unit, Cambridge, UK. URL
  • Wand, M. P., Ormerod, J. T., Padoan, S. A., and Frühwirth, R. (2011). “Mean field variational Bayes for elaborate distributions.” Bayesian Analysis, 6: 847–900.