Abstract
Consider a family \({(X_i)_{i \in I}}\) of random variables endowed with the structure of a Bayesian network, and a subset S of I. This paper examines the problem of computing the probability distribution of the subfamily \({(X_{a})_{a \in S}}\) (respectively the probability distribution of \({ (X_{b})_{b \in {\bar{S}}}}\) , where \({{\bar{S}} = I - S}\) , conditional on \({(X_{a})_{a \in S}}\)). This paper presents some theoretical results that makes it possible to compute joint and conditional probabilities over a subset of variables by computing over separate components. In other words, it is demonstrated that it is possible to decompose this task into several parallel computations, each related to a subset of S (respectively of \({{\bar{S}}}\)); these partial results are then put together as a final product. In computing the probability distribution over \({(X_a)_{a \in S}}\) , this procedure results in the production of a structure of level two Bayesian network structure for S.
Similar content being viewed by others
References
Cowell RG, Philip DA, Lauritzen SL, Spiegelhalter DJ (1999) Probabilistic networks and expert systems. Springer-Verlag, New York
Dan G, Pearl J (1989) Axioms and algorithms for inferences involving conditional independence. Technical Report CSD 890031, R-119-I, Cognitive Systems Laboratory, University of California, Los Angeles
Dan G, Verma T, Pearl J (1990) D-separation: from theorems to algorithms. In Henrion M et al (eds) Uncertainty in artificial intelligence, North Holland, New York, vol 5, pp 139–148
Dechter R (1996) Bucket elimination: a unifying framework for probabilistic inference. In: Horvits E, Jensen F (eds) Proceedings of the twelthth conference on uncertainty in artificial intelligence. Portland, Oregon, pp 211–219
Hájek P, Havránek T, Jirouśek R (1992) Information processing in expert systems. CRC Press Inc., Boca Raton
Heckerman D (1999) A tutorial on learning with Bayesian networks. In: Jordan M (eds) Learning in graphical models. MIT Press, Cambridge
Jensen FV (1996) An introduction to Bayesian networks. UCL Press, London
Jensen FV, Lauritzen SL, Olesen KG (1990) Bayesian updating in causal probabilistic networks by local computations. Comput Stat Q 4: 269–282
Judea P, Verma T (1988) Influence diagrams and D-separation. UCLA Cognitive Systems Laboratory, Technical Report 880052
Lauritzen SL, Spiegelhalter DJ (1988) Local computation with probabilities on graphical structures and their application to expert systems. Proc R Stat Soc B 50(2): 157–194
Neapolitan RE (1990) Probabilistic reasoning in expert systems. Wiley, New York
Pearl J (1988) Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, San Mateo
Pearl P, Verma T (1987) The logic of representing dependencies by directed graphs. Technical report CSD 870004, R-79-II, University of California, Los Angeles
Shafer G (1996) Probabilistic expert system. CBMS-NSF Regional Conference Series in Applied Mathematics, vol 67. SIAM, Philadelphia
Spirtes P (1994) Conditional independence in directed cyclic graphical models for feedback. Technical Report CTechnical Report CMU-PHIL-54, Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA
Spirtes P, Glymour C, Scheines R (1993) Causation, prediction, and search. Lecture notes in statistics 81. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Smail, L. D-Separation and computation of probability distributions in Bayesian networks. Artif Intell Rev 31, 87 (2009). https://doi.org/10.1007/s10462-009-9128-3
Published:
DOI: https://doi.org/10.1007/s10462-009-9128-3