73,298 research outputs found
Standard imsets for undirected and chain graphical models
We derive standard imsets for undirected graphical models and chain graphical
models. Standard imsets for undirected graphical models are described in terms
of minimal triangulations for maximal prime subgraphs of the undirected graphs.
For describing standard imsets for chain graphical models, we first define a
triangulation of a chain graph. We then use the triangulation to generalize our
results for the undirected graphs to chain graphs.Comment: Published at http://dx.doi.org/10.3150/14-BEJ611 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Contraction blockers for graphs with forbidden induced paths.
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pâ„“-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
Marginal AMP Chain Graphs
We present a new family of models that is based on graphs that may have
undirected, directed and bidirected edges. We name these new models marginal
AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP
chain graph under marginalization of some of its nodes. However, MAMP chain
graphs do not only subsume AMP chain graphs but also multivariate regression
chain graphs. We describe global and pairwise Markov properties for MAMP chain
graphs and prove their equivalence for compositional graphoids. We also
characterize when two MAMP chain graphs are Markov equivalent.
For Gaussian probability distributions, we also show that every MAMP chain
graph is Markov equivalent to some directed and acyclic graph with
deterministic nodes under marginalization and conditioning on some of its
nodes. This is important because it implies that the independence model
represented by a MAMP chain graph can be accounted for by some data generating
process that is partially observed and has selection bias. Finally, we modify
MAMP chain graphs so that they are closed under marginalization for Gaussian
probability distributions. This is a desirable feature because it guarantees
parsimonious models under marginalization.Comment: Changes from v1 to v2: Discussion section got extended. Changes from
v2 to v3: New Sections 3 and 5. Changes from v3 to v4: Example 4 added to
discussion section. Changes from v4 to v5: None. Changes from v5 to v6: Some
minor and major errors have been corrected. The latter include the
definitions of descending route and pairwise separation base, and the proofs
of Theorems 5 and
Unifying Markov Properties for Graphical Models
Several types of graphs with different conditional independence
interpretations --- also known as Markov properties --- have been proposed and
used in graphical models. In this paper we unify these Markov properties by
introducing a class of graphs with four types of edges --- lines, arrows, arcs,
and dotted lines --- and a single separation criterion. We show that
independence structures defined by this class specialize to each of the
previously defined cases, when suitable subclasses of graphs are considered. In
addition, we define a pairwise Markov property for the subclass of chain mixed
graphs which includes chain graphs with the LWF interpretation, as well as
summary graphs (and consequently ancestral graphs). We prove the equivalence of
this pairwise Markov property to the global Markov property for compositional
graphoid independence models.Comment: 31 Pages, 6 figures, 1 tabl
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