Edwards, D. (1990). Hierarchical interaction models (with Discussion). J. R. Statist.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....there are fast marginalisation and estimation algorithms. 2. MIXED BAYESIAN NETWORKS Mixed discrete and continuous variable Bayesian networks are commonly, though not exclusively [6, 7] modelled with the conditional Gaussian (CG) model of Lauritzen and Wermuth [2, 8] and its close relatives [2, 9]. CG distributions are joint distributions of a discrete variable and a vector Gaussian variable and belong to the regular exponential family: p(x; y) P (x) expf Gamma 1 2 (y Gamma x ) T Sigma Gamma1 x (y Gamma x )g j2 Sigma x j 1 2 (1) The discrete variable x interacts with ....

....models. Examples of the graphs that can be defined using discrete and CG regression models are shown in figure 2. Figure 1: Examples of Mixed Bayesian Networks. Discrete nodes are shown as squares, and continuous nodes as circles. Taking inspiration from Edwards hierarchical interaction models [9], we can also specify equivalences between the means, regression parameters and covariances over some or all of the states of the discrete variable. 3. AN EM ALGORITHM FOR MIXED BAYESIAN NETWORKS The expectation maximisation method [10] is the popular technique for solving maximum likelihood and ....

D. E. Edwards, "hierarchical interaction models, " J. Roy. Satist. Soc. B, vol. 52, no. 1, pp. 3-- 20, 1990.

....Mixed Hierarchical Interaction Models by Selection Steffen L. Lauritzen 1 ABSTRACT: This note is concerned with the class of hierarchical interaction models for mixed discrete and continuous variables as defined by Edwards (1990) and modified by Lauritzen (1996) In particular it is shown that any hierarchical log linear interaction model can be generated by selection on a set of response variables in a directed Markov model over what we have termed the selection graph of the model. An inequality is established for the ....

....undirected Markov distribution obtained by conditioning on the values of the response variables in the selection graph, thus demonstrating that not all such distributions can be generated in this way. Finally it is shown that in the mixed case only hierarchical models of the type defined by Edwards (1990) can be generated by selection as above. KEYWORDS: Bayesian networks; Conditional Gaussian distribution; Covariance selection; Gaussian graphical models; Log linear interaction models; Recursive models. 1 Introduction Although the class of log linear models for contingency tables are well ....

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Edwards, D. (1990). Hierarchical interaction models (with discussion). Journal of the Royal Statistical Society, Series B, 52, 3--20 and 51--72.

....there are fast marginalisation and estimation algorithms. 2. MIXED BAYESIAN NETWORKS Mixed discrete and continuous variable Bayesian networks are commonly, though not exclusively [6, 7] modelled with the conditional Gaussian (CG) model of Lauritzen and Wermuth [2, 8] and its close relatives [2, 9]. CG distributions are joint distributions of a discrete variable and a vector Gaussian variable and belong to the regular exponential family: p(x; y) P (x) expf 1 2 (y x ) T 1 x (y x )g j2 x j 1 2 (1) The discrete variable x interacts with the continuous variable y by ....

....models. Examples of the graphs that can be defined using discrete and CG regression models are shown in figure 2. Figure 1: Examples of Mixed Bayesian Networks. Discrete nodes are shown as squares, and continuous nodes as circles. Taking inspiration from Edwards hierarchical interaction models [9], we can also specify equivalences between the means, regression parameters and covariances over some or all of the states of the discrete variable. 3. AN EM ALGORITHM FOR MIXED BAYESIAN NETWORKS The expectation maximisation method [10] is the popular technique for solving maximum likelihood and ....

D. E. Edwards, "hierarchical interaction models, " J. Roy. Satist. Soc. B, vol. 52, no. 1, pp. 3-- 20, 1990.

....to learn the parameters m from data. The mathematics of fitting parameters to a Bayesian Markov network is an extension of standard fitting procedures in statistics. Fitting algorithms exist for Bayesian networks and more general probabilistic networks in the cases of complete and missing data [74], 42] 75] 76] See Whittaker for a more extensive discussion and review of methods and theory. In the case of a Bayesian network with complete data, where the distributions at the nodes are discrete probability tables or Gaussians, fast close form solutions exist that can be computed in time ....

D. Edwards, "Hierarchical interaction models", Journal of the Royal Statistical Society B, vol. 51, no. 3, 1989.

....very close to the EM algorithm. The algorithm alternates between a T step which calculates a tilted version of the unconditional likelihood function and an M step which maximises it. The algorithm applies to mixed graphical chain models (Lauritzen Wermuth, 1989) and their generalisations (Edwards, 1990), and was developed with these in mind, but it may have applications beyond these. The algorithm has been implemented in the most recent version of the MIM software (Edwards, 2000) where it was named the ME algorithm. The name has been changed to avoid confusion with the algorithm described by ....

....The name has been changed to avoid confusion with the algorithm described by Marschner (2001) Key words: CG distributions, conditional inference, graphical models, logistic regression. 1 1 Introduction Graphical chain models were introduced by Lauritzen Wermuth (1989) and extended by Edwards (1990). Their potential for applications has been discussed by Wermuth Lauritzen (1990) Cox Wermuth (1996) and others. Their practical use has been limited by the absence of an estimation algorithm which can be easily implemented in general software for tting and analysing graphical models. This ....

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Edwards, D. (1990). Hierarchical interaction models (with Discussion). J. R. Statist.

....are identical to computations in the EM algorithm. To re ect the structure of the algorithm and the above mentioned duality, we have chosen to refer to the algorithm as the ME algorithm. The algorithm applies to mixed graphical chain models (Lauritzen and Wermuth 1989) and their generalizations (Edwards 1990), and it was developed with these as motivation, but we believe it to have potential applications beyond these. The algorithm has been implemented in the most recent version of the MIM software (Edwards 1995) Key words: CG distributions, conditional inference, graphical models, logistic ....

....The algorithm has been implemented in the most recent version of the MIM software (Edwards 1995) Key words: CG distributions, conditional inference, graphical models, logistic regression. 1 Introduction Graphical chain models were introduced by Lauritzen and Wermuth (1989) and extended by Edwards (1990). Their potential for applications has been discussed by Wermuth and Lauritzen (1990) Cox and Wermuth (1996) and others. Their practical use has been limited by the absence of an estimation algorithm which can be easily implemented in general software for tting This is Research Report ....

Edwards, D. (1990). Hierarchical interaction models (with discussion). Journal of the Royal Statistical Society, Series B, 52, 3-20 and 51-72.

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Edwards, D. (1990). Hierarchical interaction models (with discussion). Journal of the Royal Statistical Society, series B, 52(1): 3-20.

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