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Chain Graph Models and their Causal Interpretations
 B
, 2001
"... Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultim ..."
Abstract

Cited by 46 (4 self)
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Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultimately fallacious interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to awed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated as the equilibrium distribution of dynamic models with feedback. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for DAGs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feedback in econometrics. Keywords: Causal model; cha...
The Multiinformation Function As A Tool For Measuring Stochastic Dependence
 Learning in Graphical Models
, 1998
"... . Given a collection of random variables [¸ i ] i2N where N is a finite nonempty set, the corresponding multiinformation function ascribes the relative entropy of the joint distribution of [¸ i ] i2A with respect to the product of distributions of individual random variables ¸ i for i 2 A to every s ..."
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Cited by 32 (0 self)
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. Given a collection of random variables [¸ i ] i2N where N is a finite nonempty set, the corresponding multiinformation function ascribes the relative entropy of the joint distribution of [¸ i ] i2A with respect to the product of distributions of individual random variables ¸ i for i 2 A to every subset A ae N . We argue it is a useful tool for problems concerning stochastic (conditional) dependence and independence (at least in discrete case). First, it makes possible to express the conditional mutual information between [¸ i ] i2A and [¸ i ] i2B given [¸ i ] i2C (for every disjoint A; B; C ae N) which can be considered as a good measure of conditional stochastic dependence. Second, one can introduce reasonable measures of dependence of level r among variables [¸ i ] i2A (where A ae N , 1 r ! card A) which are expressible by means of the multiinformation function. Third, it enables one to derive theoretical results on (nonexistence of an) axiomatic characterization of stochastic c...
Factorization of Discrete Probability Distributions
 UAI 2002
, 2002
"... We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. This result generalizes the well known HammersleyClifford Theorem. ..."
Abstract
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We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a loglinear model, or other more general exponential models. This result generalizes the well known HammersleyClifford Theorem.