Geiger D, Heckerman D. A characterization of the Dirichlet distribution with application to learning Bayesian networks. In: Besnard P, Hanks S, editors. Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence (UAI-1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....not manipulated in other cases. In order to efficiently evaluate the integral in Equation 6, researchers have introduced several assumptions that lead to a closed form solution (Cooper and Herskovits 1992, Heckerman, et al. 1995) Under the assumptions that follow in this section, as expressed in (Geiger and Heckerman 1995), the integral in Equation 6 can be computed efficiently in closed form. Assumption 5. Variables are discrete. Assumption 6 (parameter independence) Global parameter independence: For each causal Bayesian network structure, the parameters (probabilities) associated with one node are ....

Geiger, D. and Heckerman, D. (1995) A characterization of the Dirichlet distribution with application to learning Bayesian networks, In: Proceedings of the Conference on Uncertainty in Artificial Intelligence 196-207.

....modeling prior knowledge in the Bayesian analysis of categorical data, more general priors, such as mixtures of Dirichlet distributions, sometimes may be needed to adequately reflect prior knowledge (Bernardo and Smith (1994) p. 279) When working in the space of individual ADG models, however, Geiger and Heckerman (1995) show that the Dirichlet family is the only family of prior distributions that can be used to achieve score equivalence. Working in the space of Markov equivalence classes, conveniently represented by essential graphs, eliminates the issue of score equivalence and therefore allows the adoption of ....

Geiger, D. and D. Heckerman (1995). A characterization of the Dirichlet distribution with applications to learning Bayesian networks. In Uncertainty in Artificial Intelligence, Proceedings of the Eleventh Conference (P.Besnard and S. Hanks, eds.), San Francisco: Morgan Kaufman, pp.

.... Gaussians: SK89] AHSE93] DM95] mixture) CF91] Lau92] GH94b] CG Distributions: LW89] Ole93] Exponential Families: Bun95a] Whi90] Issues of Conjugacy: Dawid s response to [LS88] Continuous Belief Function Densities: WD94b] WD94a] others: AFS94] KSC84] GH94a] GH95a] Ken86] 1.10 Philosophical Issues 1.10.1 Causality and Control refs: Pea88a] Pea94b] Pea94a] Pea95b] Pea95a] SGS93] DS93] HS94] CS92] BP94] BP95] GP95] GP96] HB94b] Pea96] Sto93] 1.10.2 Modeling Various critiques and or criticisms on the problems and issues ....

....Influence diagram: LS93] 3 Learning Acquiring Models Survey: Bun95a] Hec95b] 3.1 Learning probabilities given structure: General purpose EM techniques not listed) LPP95] Mus93] 3. 2 Learning structure Bayesian Learning Approaches: HGC94, HGC95] HG95] CGH94] Hec95b] Hec95a] GH95a] GH95b] HB94a] DL93b] CH91b] CH91a] AC94] Bad92] Bun94] CH92] Coo95b] Coo95a] SDLC93] Bou94] Bun91] Bun95b] GH94b] MDL: LB93] Suz93] Non Bayesian Learning Approaches: These usually involve statistical tests to find conditional independencies, followed by searches ....

Dan Geiger and David Heckerman. A characterization of the Dirichlet distribution with application to learning Bayesian networks. In Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence, Montreal, August 1995. Morgan Kauffman.

....from data and network equivalence are a precursor to techniques for learning from medium sized samples of Fig. 7. Network equivalence is an important concept used in some Bayesian techniques for learning Bayesian networks from data, used in advanced work on priors for Bayesian networks [105], 37] This will be discussed later. VI. Diagnostics, elicitation and assessment The day to day practice of learning and data analysis may have a learning algorithm at its core but a lot of the work involves modeling and assessment: building a model and trying to find out what is going on with ....

....out in one form or another, by many [140] 35] 121] 111] 112] 68] 141] 142] 143] 37] 38] Certainly, these techniques use standard Bayesian manipulations and should be obvious to most students of Bayesian theory. The general case for the exponential family is worked through in [105]. Good summaries of this line of work can be found in [111] 68] 144] 37] 23] and a thesis covering many of the issues is [36] The full Bayesian approach is a predictive one: rather than returning the single best network, the aim might be to perform prediction or estimate probabilities ....

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D. Geiger and D. Heckerman, "A characterization of the Dirichlet distribution with application to learning Bayesian networks ", In Besnard and Hanks [158].

....cases in D in which X i = x k i and Pa i = pa j i . We call this expression the Cooper Herskovits scoring function. The last two assumptions are made for the sake of convenience. Namely, the parameter distributions before and after data are seen are in the same family: the Dirichlet family. Geiger and Heckerman (1995) provide a characterization of the Dirichlet distribution, which shows that the fifth assumption is implied from the first three assumptions and from one additional assumption that if S 1 and S 2 are equivalent Bayesian networks (i.e. they represent the same sets of joint distributions) then the ....

Geiger, D. and Heckerman, D. (1995). A characterization of the Dirichlet distribution with application to learning Bayesian networks. In Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, pages 196--207. Morgan Kaufmann.

....we discuss an extension of our characterization from two ways tables to n way tables and the implication to learning Bayesian networks. Further extensions are described in Section 4. An analogous result for a normal sampling situation which characterizes the Wishart distribution is outlined in [GH95]. 2 Background and Technical Summary The Dirichlet pdf is defined as follows. Let OE 1 ; OE l be positive random variables that sum to 1. Then OE 1 ; OE l Gamma1 have a Dirichlet pdf f if f(OE 1 ; OE l Gamma1 ) Gamma( P l i=1 ff i ) Q l i=1 Gamma(ff i ) l Y i=1 ....

D. Geiger and D. Heckerman, A characterization of the Dirichlet distribution with application to learning Bayesian networks. Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, 196--207, August 1995. Morgan Kaufmann.

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Geiger D, Heckerman D. A characterization of the Dirichlet distribution with application to learning Bayesian networks. In: Besnard P, Hanks S, editors. Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence (UAI-1995.

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