D. Geiger and D. Heckerman. A characterization of the dirichlet distribution through global and local parameter independence. The Annals of Statistics, 25:1344--1369, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....random individual is of class k given conditions xi. 2. 3 The prior distributions Priors for bik s: The natural distribution of random variables on a simplex with a strictly positive density, that satisfy certain natural mutual conditional indepen dence assumptions, is the Dirichlet distribution [GHe97]. Consequently, we assume that bik s are Dirichlet distributed ( ilk, bill, M) Di(kl, kM) 10) where ni s are global hyperparameters representing general evidence for the response model i. A non informative (uniform) choice is to set ni : 1 for all 1 i M. The Dirichlet ....

Geiger, D. & Heckerman, D., Characterization of the Dirichlet distribu- tion through global and local parameter independence, Annals of Statistics 25:1344-1369, 1997.

....modelling 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 through local and global independence. In Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence, Philippe Besnard and Steve Hanks, eds., pp. 196-207. Morgan Kaufmann, San Mateo.

....ij , ff ij = P c i k=1 ff ijk is the prior precision on ij , and ff = P ij ff ij is the size of the imaginary database. The situation of initial ignorance can be represented by assuming ff ijk = ff= c i q i ) for all i, j and k, so that the prior probability of (x ik j ij ) is simply 1=c i [5]. 15] shows that parameter independence and prior Dirichlet distributions imply that the posterior density of is still a product of Dirichlet densities and ij jD D(ff ij1 n(x i1 j ij ) ff ijc i n(x ic i j ij ) so that local and global independence are retained after the ....

D Geiger and D Heckerman. A characterization of Dirichlet distributions through local and global independence. Ann. Statist., 25:1344--1368, 1997.

....modelling 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 12 Since f (cl D (t) is global [ D ) c l D (t) m ....

Geiger, D. and D. Heckerman (1995). A characterization of the Dirichlet distribution through local and global independence. Submitted for publication.

....and global independence (Spiegelhalter and Lauritzen, 1990) in the field of Bayesian Belief Networks. We merely note here that, as long as the hyperparameters of the prior distributions are chosen consistently, exact Bayesian analysis leads to identical conclusions under both parameterizations (Geiger and Heckerman, 1997). 2.2 Incomplete Samples Exact Bayesian analysis presents no difficulties as long as the sample is complete. Suppose now that some of the entries on the variable Y are reported as unknown. Let S = S o ; Sm ) where S o and Sm respectively denote the sample with complete observations and the one ....

Geiger, D., and Heckerman, D. (1997). A characterization of Dirichlet distributions through local and global independence. Ann. Statist., 25, 1344--1368.

.... on network parameters, and some other assumptions [2] It was later shown that the assumption of global and local parameter independence for all nodes in every complete network structure dictates that the only possible prior parameter distribution for discrete DAG models is a Dirichlet prior [5, 7]. In contrast, in a subsequent work, it was shown that for Gaussian DAG models, which consist of a recursive set of linear regression models, global independence alone dictates that the only feasible parameter prior is the Normal Wishart distribution, assuming models with at least three nodes ....

....g) J Gamma1 1 f 1 (f i Delta g)g 1 (f jji g) p(f ij g) J Gamma1 2 f 2 (f Deltaj g)g 2 (f ijj g) 1) where J 1 ,J 2 are appropriate Jacobians and Deltaj , ijj are defined similarly to i Delta and jji . We formulate the following theorem that extends the result stated in [5] for two node DAG models with binary variables. Theorem 1 Any probability distribution on f ij g that satisfies the regularity assumption (1) and global parameter independence assumption (Equation 1) is of the form p(f ij g) C h Q n i=1 Q k j=1 ff ij ij i Delta H in ij ....

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D. Geiger and D. Heckerman. A characterization of the dirichlet distribution through global and local parameter independence. The Annals of Statistics, 25(3):1344--1369, 1997.

....Theorems 6, 8 and 9, respectively, proven in Section 5. It should be noted that a single principle, global parameter independence, is used to characterizes three different distributions. In Section 6, we compare these characterizations to a recent characterization of the Dirichlet distribution (Geiger and Heckerman, 1997; J arai , 1998) and conjecture that the later characterization uses a redundant assumption (local parameter independence) that is, global parameter independence may also characterize the Dirichlet distribution. The Dirichlet, normal, Wishart, and normal Wishart distributions are the conjugate ....

....formula for the marginal likelihood applies whenever Assumptions 1 through 5 are satisfied, not only for Gaussian DAG models. Another important special case is when all variables in X are discrete and all local distributions are multinomial. This case has been treated in (Heckerman et al. 1995; Geiger and Heckerman, 1997) under the additional assumption of local parameter independence. Our generalized derivation herein dispenses this assumption and unifies the derivation in the discrete case with the derivation needed for Gaussian DAG models. Furthermore, our proof also suggests that the only parameter prior for ....

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Geiger, D. and Heckerman, D. (1997). A characterization of the Dirichlet distribution through global and local parameter independence. Annals of Statistics, 25(3):1344--1369.

....functional equations [Ac66] In section 2 we develop the needed functional equation and use results of [Ja86] to prove that every positive measurable solution of it has infinitely many derivatives. In sections 3 through 5 we gradually solve this equation and in Section 6 we outline a related work [GH95] and possible extensions along with their statistical application. 2 The functional equation Let x = fx 1 ; x 2 g have a non singular Bivariate normal pdf f( x) N( W ) If we write f( x) f(x 1 )f(x 2 jx 1 ) where f(x 1 ) N(m 1 ; 1=v 1 ) and f(x 2 jx 1 ) N(m 2j1 b 12 x 1 ; 1=v 2j1 ....

....of positiveness is actually redundant. The importance of J arai s results is that in the process of solving Equation 7 one may take as many derivatives as is found needed. Further examples of the applicability of J arai s results to characterization problems in statistics are discussed in [GH95] along with an example where they fail. 3 The Bivariate equation with a fixed precision matrix We shall first assume that the variances v 1 and v 2j1 in Equation 7 are fixed and that b 12 is fixed and is not zero. Thus, by renaming variables and functions, Equation 7 can be stated as follows: ....

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D. Geiger and D. Heckerman, A characterization of the Dirichlet distribution through local and global parameter independence. Submitted to Annals of Statistics, February 1995.

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D. Geiger and D. Heckerman. A characterization of the dirichlet distribution through global and local parameter independence. The Annals of Statistics, 25:1344--1369, 1997.

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Geiger, D. and D. Heckerman: 1994, `A Characterization of the Dirichlet Distribution Through Global and Local Independence'. Technical Report MSR-TR-94-16, Microsoft Research.

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Geiger, D. and D. Heckerman: 1994, `A Characterization of the Dirichlet Distribution Through Global and Local Independence'. Technical Report MSR-TR-94-16, Microsoft Research.

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D. Geiger and D. Heckerman, A characterization of the Dirichlet distribution through global and local independence, Computer Science Department, Technical report 9506, February 1995. A preliminary report appears as Microsoft Research Report, TR-94-16.

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