Access the full text.
Sign up today, get DeepDyve free for 14 days.
(1967)
Lattice Theory, American Mathematical Society
A. Dawid (1979)
Conditional Independence in Statistical TheoryJournal of the royal statistical society series b-methodological, 41
S. Lauritzen, N. Wermuth (1989)
Graphical Models for Associations between Variables, some of which are Qualitative and some QuantitativeAnnals of Statistics, 17
J. Darroch, S. Lauritzen, T. Speed (1980)
Markov Fields and Log-Linear Interaction Models for Contingency TablesAnnals of Statistics, 8
D. Geiger, J. Pearl (2013)
On the logic of causal models
(1988)
Local computations on graphical structures and their application to expert systems
(1988)
LISREL 7: A guide to the program and applications
S. Andersson, M. Perlman (1995)
Testing lattice conditional independence modelsJournal of Multivariate Analysis, 53
J. Besag (1974)
Spatial Interaction and the Statistical Analysis of Lattice SystemsJournal of the royal statistical society series b-methodological, 36
S. Andersson, D. Madigan, M. Perlman, C. Triggs (1995)
On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphsJournal of Statistical Planning and Inference, 48
(1979)
A note on nearest-neighbour Gibbs and Markov probabilities
S. Andersson, M. Perlman (1993)
Lattice Models for Conditional Independence in a Multivariate Normal DistributionAnnals of Statistics, 21
N. Wermuth (1980)
Linear Recursive Equations, Covariance Selection, and Path AnalysisJournal of the American Statistical Association, 75
(1995)
Directed cyclic graphical representation of feedback
(1989)
Sörbom, LISREL 7. A Guide to the Program and Applications, 2nd edition (Scientific
M. Studený (1993)
Formal Properties of Conditional Independence in Different Calculi of AI
S. Lauritzen, A. Dawid, B. Larsen, H.-G. Leimer (1990)
Independence properties of directed markov fieldsNetworks, 20
P. Spirtes (1995)
Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence
S. Andersson, D. Madigan, M. Perlman, C. Triggs (1997)
A graphical characterization of lattice conditional independence modelsAnnals of Mathematics and Artificial Intelligence, 21
(1984)
Recursive causal models, Journal of the Australian Mathematical Society, Series A
S. Wright (1934)
The Method of Path CoefficientsAnnals of Mathematical Statistics, 5
P. Spirtes (1995)
Directed Cyclic Graphical Representations of Feedback Models
M. Studený, R. Bouckaert (1998)
On chain graph models for description of conditional independence structuresAnnals of Statistics, 26
S. Andersson, D. Madigan, M. Perlman (1997)
On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic DigraphsScandinavian Journal of Statistics, 24
J. Pearl (1986)
Uncertainty in Artificial Intelligence
J. Pearl (1991)
Probabilistic reasoning in intelligent systems - networks of plausible inference
J. Pearl (1985)
A Constraint-Propagation Approach to Probabilistic Reasoning
M. Frydenberg (1990)
The chain graph Markov propertyScandinavian Journal of Statistics, 17
H. Kiiveri, T. Speed, J. Carlin (1984)
Recursive causal modelsJournal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 36
(1990)
Leimer, Independence properties of directed Markov fields, Networks
BY Koster (1996)
Markov properties of nonrecursive causal modelsAnnals of Statistics, 24
Steffen Lauritzen, J. Whittaker (1990)
Graphical Models in Applied Multivariate Statistics.The Statistician, 42
Directed acyclic graphs (DAG's) and, more generally, chain graphs have in recent years been widely used for statistical modelling. Their Gibbs and Markov properties are now well understood and are exploited, e.g., in reducing the complexity encountered in estimating the joint distribution of many random variables. The scope of the models has been restricted to acyclic or recursive processes and this restriction was long considered imperative, due to the supposed fundamentally different nature of processes involving reciprocal interactions between variables. Recently however it was shown independently by Spirtes (Spirtes, 1995) and Koster (Koster, 1996) that graphs containing directed cycles may be given a proper Markov interpretation. This paper further generalizes the scope of graphical models. It studies a class of conditional independence (CI) probability models determined by a general graph which may have directed and undirected edges, and may contain directed cycles. This class of graphical models strictly includes the well-known class of graphical chain models studied by Frydenberg et al., and the class of probability models determined by a directed cyclic graph or a reciprocal graph, studied recently by Spirtes and Koster. It is shown that the Markov property determined by a graph is equivalent to the existence of a Gibbs-factorization of the density (assumed positive). To better understand the structural aspects of the Gibbs and Markov properties embodied by graphs the notion of lattice conditional independence (LCI), introduced by Andersson and Perlman (Andersson and Perlman, 1993), is needed. The Gibbs-factorization has an outer ‘skeleton’ which is determined by the ring of all anterior sets of the graph.
Annals of Mathematics and Artificial Intelligence – Springer Journals
Published: Sep 29, 2004
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.