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Bayesian networks with a logistic regression model for the conditional probabilities. (English) Zbl 1184.62039
Summary: Logistic regression techniques can be used to restrict the conditional probabilities of a Bayesian network for discrete variables. More specifically, each variable of the network can be modeled through a logistic regression model, in which the parents of the variable define the covariates. When all main effects and interactions between the parent variables are incorporated as covariates, the conditional probabilities are estimated without restrictions, as in a traditional Bayesian network. By incorporating interaction terms up to a specific order only, the number of parameters can be drastically reduced. Furthermore, ordered logistic regression can be used when the categories of a variable are ordered, resulting in even more parsimonious models. Parameters are estimated by a modified junction tree algorithm. The approach is illustrated with an Alarm network.

##### MSC:
 62F15 Bayesian inference 62J12 Generalized linear models (logistic models) 65C60 Computational problems in statistics (MSC2010)
Fahrmeir
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##### References:
 [1] Albert, A.; Anderson, J.A., On the existence of maximum likelihood estimates in logistic regression models, Biometrika, 71, 1-10, (1984) · Zbl 0543.62020 [2] Buntine, W.L., A guide to the literature on learning probabilistic networks from data, IEEE transactions on knowledge and data engineering, 8, 195-210, (1996) [3] Edwards, A.; Thurstone, L., An internal consistency check for scale values determined by the method of successive intervals, Psychometrika, 17, 169-180, (1952) [4] Fahrmeir, L.; Tutz, G., Multivariate statistical modelling based on generalized linear models, (2000), Springer-Verlag New York [5] Friedman, N.; Goldszmidt, M., Learning Bayesian networks with local structure, () · Zbl 0910.68176 [6] E.H. Herskovits, Computer-based probabilistic-network construction. PhD Thesis. Stanford University, 1991. [7] Jensen, F.V.; Lauritzen, S.L.; Olesen, K.G., Bayesian updating in causal probabilistic networks by local computation, Computational statistics quarterly, 4, 269-282, (1990) · Zbl 0715.68076 [8] () [9] Lauritzen, S.L., The EM algorithm for graphical association models with missing data, Computational statistics and data analysis, 19, 191-201, (1995) · Zbl 0875.62237 [10] Little, R.J.A.; Rubin, D.B., Statistical analysis with missing data, (1987), Wiley New York [11] Neal, R.M., Connectionist learning of belief networks, Artificial intelligence, 56, 71-113, (1992) · Zbl 0761.68081 [12] Neapolitan, R.E., Learning Bayesian networks, (2004), Pearson Prentice Hall [13] Saul, L.K.; Jaakkola, T.S.; Jordan, M.I., Mean field theory for Sigmoid belief networks, Journal of artificial intelligence research, 4, 61-76, (1996) · Zbl 0900.68379
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