Subgraph Approximations for Directed Graphical Models
Graphical Models provide a powerful tool for the formulation of general statistical models. In a previous paper, (Yiannoutsos and Gelfand, 1991), the authors argued that sampling-based techniques provide a unified approach for the analysis of graphical models under general distributional specifications. These techniques include both noniterative and iterative Monte Carlo.
Our concern here is with moderate to large possibly highly connected graphical models whose size and complexity may prohibit analysis within a reasonable time frame. Typically in large systems however, interest focuses on the behavior of only a few critical nodes. Our proposal is to develop, for a particular node, an approximating subgraph which contains virtually as much information about the variable as the full network, but by virtue of its reduced size, enables rapid computational investigation. We provide an illustration using a 40-node graph. Though this is not as large as we would envision in practice, it is convenient in permitting full model calculations to enable assessment of our approximations.
KeywordsSource Node Graphical Model Conditional Distribution Linear Chain Critical Node
Unable to display preview. Download preview PDF.
- DeGroot, M. H. and Goel, P. K. (1986) Information and Bayesian hierarchical models. Unpublished Manuscript.Google Scholar
- Devroye, L. (1986) Non-Uniform Random Number Generation. Springer-Verlag, New York.Google Scholar
- Henrion, M. (1988). Propagating uncertainty in Bayesian networks by probabilistic logic sampling. In Uncertainty in Artificial Intelligence, Vol. 2, J. Lemmer, and L. N. Kanal (Eds.). North-Holland: Amsterdam, 149–164.Google Scholar
- Lauritzen, S. L. (1990a). Propagation of probabilities, means and variances in mixed graphical association models. Research Report 90–18, Aalborg University.Google Scholar
- Muller, P. (1992). A generic approach to posterior integration and Gibbs sampling. Journal of American Statistical Association, (to appear).Google Scholar
- Oh, M.-S., and Berger, J. O. (1992). Adaptive imputation sampling in Monte Carlo integration. Journal of Statistical Computing and Simulation, (to appear).Google Scholar
- Rubin, D. (1988). Using the SIR algorithm to simulate posterior distributions. In Bayesian Statistics 3, Eds. J. Bernardo et al. Oxford University Press, 395–402.Google Scholar
- Shachter, R. D. and Peot, M. A. (1989). Evidential reasoning using likelihood weighting. Artificial Intelligence, (submitted).Google Scholar
- Tierney, L. (1991). Markov chains for exploring posterior distributions. Technical Report No. 560, School of Statistics, University of Minnesota.Google Scholar
- West, M. (1992). Approximating posterior distributions by mixtures. In Bayesian Statistics 4, Eds. J. Bernardo et al. Oxford University Press, 505–524.Google Scholar
- Yiannoutsos, C. T. and Gelfand, A. E. (1991). Simulation approaches for calculations in directed graphical models. Technical Report No. 91–23, Department of Statistics, University of Connecticut.Google Scholar