Bayesian Analysis

Dynamic matrix-variate graphical models

Carlos M. Carvalho and Mike West

Full-text: Open access


This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce structured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We define this new class of models and their theoretical structure involving novel matrix-normal/hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and computational strategies for model fitting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full sequential and conjugate updating, filtering and forecasting analysis. The models are then applied in the context of financial time series for predictive portfolio analysis. The improvements defined in optimal Bayesian decision analysis in this example context vividly illustrate the practical benefits of the parsimony induced via appropriate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.

Article information

Bayesian Anal., Volume 2, Number 1 (2007), 69-97.

First available in Project Euclid: 22 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Database Expansion Item

Bayesian Forecasting Dynamic Linear Models Gaussian Graphical Models Graphical Model Uncertainty Hyper-Inverse Wishart Distribution Portfolio Analysis


Carvalho, Carlos M.; West, Mike. Dynamic matrix-variate graphical models. Bayesian Anal. 2 (2007), no. 1, 69--97. doi:10.1214/07-BA204.

Export citation


  • Aguilar, O. and West, M. (2000). "Bayesian dynamic factor models and variance matrix discounting for portfolio allocation." Journal of Business and Economic Statistics, 18: 338–357.
  • Bollerslev, T., Chou, R., and Kroner, K. (1992). "$\mbox{ARCH}$ modeling in finance." Journal of Econometrics, 52: 5–59.
  • Calder, C., Lavine, M., Muller, P., and Clark, J. (2003). "Incorporating multiple sources of stochasticity into dynamic population models." Ecology, 84: 1395–1402.
  • Carvalho, C., Massam, H., and West, M. (2005). "Simulation of hyper-inverse Wishart distributions in graphical models." ISDS Discussion Paper.
  • Dawid, A. P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application." Biometrika, 68: 265–274.
  • Dawid, A. P. and Lauritzen, S. L. (1993). "Hyper-$\mbox{M}$arkov laws in the statistical analysis of decomposable graphical models." The Annals of Statistics, 3: 1272–1317.
  • Fong, W., Godsill, S. J., Doucet, A., and West, M. (2002). "Monte Carlo smoothing with application to speech enhancement." IEEE Trans. Signal Processing, 50: 438–449.
  • Godsill, S. J., Doucet, A., and West, M. (2004). "Monte Carlo smoothing for non-linear time series." Journal of the American Statistical Association, 99: 156–168.
  • Godsill, S. J. and Rayner, P. J. W. (1998). Digital Audio Restoration: A Statistical Model-Based Approach. Springer-Verlag.
  • Jacquier, E., Polson, N., and Rossi, P. (1994). "Bayesian Analysis of stochastic volatility models." Journal of Business and Economic Statistics, 12: 371–417.
  • Jones, B., Carvalho, C., Dobra, A., Hans, C., Carter, C., and West, M. (2005). "Experiments in stochastic computation for high-dimensional graphical models." Statistical Science, 20: 388–400.
  • Kim, S., Shephard, N., and Chib, S. (1998). "Stochastic volatility: likelihood inference and comparison with ARCH" model. Review of Economic Studies, 65: 361–393.
  • Lauritzen, S. L. (1996). Graphical Models. Clarendon Press, Oxford.
  • Ledoit, O. and Wolf, M. (2004). "Honey, $\mbox{I}$ shrunk the sample covariance matrix." Journal of Portfolio Management, 30: 110–119.
  • Liu, J. (2000). Bayesian Time Series Analysis: Methods Using Simulation-Based Computation. Duke University: PhD. Thesis.
  • Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. New York, USA: John Wiley and Sons.
  • Perold, A. (1988). "Large-scale portfolio optimization." Management Science, 30: 1143–1160.
  • Putnam, B. and Quintana, J. (1994). "New $\mbox{B}$ayesian statistical approaches to estimating and evaluating models of exchange rates determination." In Proceedings of the ASA Section on Bayesian Statistical Science. American Statistical Association.
  • Quintana, J. (1987). Multivariate Bayesian Forecasting Models. University of Warwick: PhD. Thesis.
  • –- (1992). "Optimal Portfolios of forward currency contracts." In Berger, J., Bernardo, J., Dawid, A., and Smith, A. (eds.), Bayesian Statistics IV, 753–762. Oxford University Press.
  • Quintana, J., Chopra, V., and Putnam, B. (1995). "Global Asset Alocation: Stretching returns by shrinking forecasts." In Proceedings of the ASA Section on Bayesian Statistical Science. American Statistical Association.
  • Quintana, J., Lourdes, V., Aguilar, O., and Liu, J. (2003). "Global gambling." In Bernardo, J., Bayarri, M., Berger, J., Dawid, A., Heckerman, D., Smith, A., and West, M. (eds.), Bayesian Statistics VII, 349–368. Oxford University Press.
  • Quintana, J. and Putnam, B. (1996). "Debating currency markets efficiency using multiple-factor models." In Proceedings of the ASA Section on Bayesian Statistical Science. American Statistical Association.
  • Quintana, J. and West, M. (1987). "Multivariate time series analysis: $\mbox{N}$ew techniques applied to international exchange rate data." The Statistician, 36: 275–281.
  • Roverato, A. (2000). "Cholesky decomposition of a hyper-inverse Wishart matrix." Biometrika, 87: 99–112.
  • Stevens, G. (1998). "On the inverse of the covariance matrix in portfolio analysis." The Journal of Finance, 53: 1821–1827.
  • Uhlig, H. (1994). "On singular Wishart and singular multivariate beta distributions." Annals of Statistics, 22: 395–405.
  • West, M. and Harrison, P. (1997). Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.
  • West, M., Prado, R., and Krystal, A. (1999). "Evaluation and comparison of EEG traces: Latent Structure in Non-Stationary Time Series." Journal of the American Statistical Association, 94: 1083–1095.
  • Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Chichester, United Kingdom: John Wiley and Sons.