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Parameter priors for directed acyclic graphical models and the characterization of several probability distributions
 MICROSOFT RESEARCH, ADVANCED TECHNOLOGY DIVISION
, 1999
"... We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normalWishart distribution. Our analysis is based on the following new characterization of the Wishart distri ..."
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Cited by 36 (1 self)
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We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normalWishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n × n, n ≥ 3, positivedefinite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W11 − W12W −1 is independent 22 W ′ 12 of {W12, W22} for every block partitioning
Local Computation with Valuations from a Commutative Semigroup
 Annals of Mathematics and Artificial Intelligence
, 1996
"... This paper studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGINlike architecture (Jensen et al. 1990) or in the architecture of Lauritzen and Spiegelhalter (1988). In particul ..."
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Cited by 33 (9 self)
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This paper studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGINlike architecture (Jensen et al. 1990) or in the architecture of Lauritzen and Spiegelhalter (1988). In particular it is shown that propagation of belief functions can be performed in these architectures. Keywords: articial intelligence, belief function, constraint propagation, expert system, probability propagation, valuationbased system. 1 Introduction An important development in articial intelligence is associated with an abstract theory of local computation known as the Shafer{Shenoy axioms (Shafer and Shenoy 1988; Shenoy and Shafer 1990). These describe in a very general setting how computations can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even though the axioms were developed to formalize computation with belief functions (Shaf...
Probabilistic expert systems for forensic inference from genetic markers
 Scandinavian Journal of Statistics
, 2002
"... ABSTRACT. We present a number of real and fictitious examples in illustration of a new approach to analysing complex cases of forensic identification inference. This is effected by careful restructuring of the relevant pedigrees as a Probabilistic Expert System. Existing software can then be used to ..."
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Cited by 23 (7 self)
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ABSTRACT. We present a number of real and fictitious examples in illustration of a new approach to analysing complex cases of forensic identification inference. This is effected by careful restructuring of the relevant pedigrees as a Probabilistic Expert System. Existing software can then be used to perform the required inferential calculations. Specific complications which are readily handled by this approach include missing data on one or more relevant individuals, and genetic mutation. The method is particularly valuable for disputed paternity cases, but applies also to certain criminal cases.
Normal Linear Regression Models with Recursive Graphical Markov Structure
 J. MULTIVARIATE ANAL
, 1998
"... A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended ..."
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Cited by 18 (6 self)
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A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended in anatur way tonor linear rear models whose LFs continue to admit suchr efactorr frr which maximum likelihoodestimator and likelihoodr (LR) test statistics can beder ed by classical linear methods. The centrdistr  of the LR test statisticfor testing one such multivariv norv linear rear model against another isder ed, and there of theseresesion models to blockr enor linear systems is established. It is shown how a collection of nonnested dependentnor linear rear models (# seemingly unringly ringly can be combined into a single multivariv norvlinear rn grear model by imposing apar set of graphical Markov (# conditional independence) restrictions.
The Size Distribution for Markov Equivalence Classes of Acyclic Digraph Models
, 2001
"... Bayesian networks, equivalently graphical Markov models determined by acyclic digraphs or ADGs (also called directed acyclic graphs or dags), have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations. However, model search ..."
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Cited by 16 (1 self)
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Bayesian networks, equivalently graphical Markov models determined by acyclic digraphs or ADGs (also called directed acyclic graphs or dags), have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations. However, model search and selection is potentially complicated by the manytoone correspondence between ADGs and the statistical models that they represent. If the ADGs/models ratio is large, search procedures based on unique graphical representations of equivalence classes of ADGs could provide substantial computational efficiency. Hitherto, the value of the ADGs/models ratio has been calculated only for graphs with n=5 or fewer vertices. In the present study, a computer program was written to enumerate the equivalence classes of ADG models and study the distributions of class sizes and number of edges for graphs up to n=10 vertices. The ratio of ADGs to numbers of classes appears to approach an asymptote of about 3.7. Distributions of the classes according to number of edges and class size were produced which also appear to be approaching asymptotic limits. Imposing a bound on the maximum number of parents to any vertex causes little change if the bound is sufficiently large, with four being a possible minimum. The program also includes a new variation of orderly algorithm for generating undirected graphs.
Separation An Completeness Properties For Amp Chain Graph Markov Models
 Ann. Statist
, 2000
"... This paper introduces ..."
A graphical characterization of lattice conditional independence models
 Ann. Math. and Artificial Intelligence
, 1997
"... Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coin ..."
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Cited by 13 (2 self)
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Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nonmonotone missing data patterns and of nonnested dependent linear regression models ( ≡ seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graphtheoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (exponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities. 1. Introduction. The use of directed graphs to represent possible dependencies among statistical variables dates back to Wright (1921) and has generated considerable research activity in the social and natural sciences. Since 1980, particular attention has been directed at
Multiscale Graphical Modeling in Space: Applications to Command and Control
, 2000
"... Recently, a class of multiscale treestructured models was introduced in terms of scalerecursive dynamics defined on trees. The main advantage of these models is their association with a fast, recursive, Kalmanfilter prediction algorithm. In this article, we propose a more general class of multisca ..."
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Cited by 11 (1 self)
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Recently, a class of multiscale treestructured models was introduced in terms of scalerecursive dynamics defined on trees. The main advantage of these models is their association with a fast, recursive, Kalmanfilter prediction algorithm. In this article, we propose a more general class of multiscale graphical models over acyclic directed graphs, for use in command and control problems. Moreover, we derive the generalizedKalmanfilter algorithm for graphical Markov models, which can be used to obtain the optimal predictors and prediction variances for multiscale graphical models. 1 Introduction Almost every aspect of command and control (C2) involves dealing with information in the presence of uncertainty. Since information in a battlefield is never precise, its status is rarely known exactly. In the face of this uncertainty, commanders must make decisions, issue orders, and monitor the consequences. The uncertainty may come from noisy data or, indeed, regions of the battle space whe...
Generic reversible jump MCMC using graphical models
, 2005
"... field of Bayesian statistics. Their enormous power and their generalizability have rendered them the method of choice for statistical inference in many scientific disciplines. Their power is so great that they can even accommodate situations in which the structure of the statistical model itself is ..."
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Cited by 8 (1 self)
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field of Bayesian statistics. Their enormous power and their generalizability have rendered them the method of choice for statistical inference in many scientific disciplines. Their power is so great that they can even accommodate situations in which the structure of the statistical model itself is uncertain. However, the analysis of such “transdimensional ” models is not easy, with several significant technical and practical difficulties to overcome. In this paper we present a class of graphical models that allow relatively straightforward analysis of a subset of these transdimensional problems. We also present a ‘guided tour ’ of the reversible jump methodology underlying our approach and discuss how each of the various difficulties has been circumvented. Our approach has been implemented using the WinBUGS framework as a GibbsMetropolis sampling ‘engine’. The main advantage of this is that it affords the analyst much modelling flexibility: transdimensional subgraphs may be used as generic components within an arbitrarily wide range of Bayesian graphical models. We present three example analyses to illustrate our approach.
The Minimum Description Length Principle and NonDeductive Inference
 PROCEEDINGS OF THE IJCAI WORKSHOP ON ABDUCTION AND INDUCTION IN
, 1997
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