39 research outputs found
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
Markov properties for mixed graphs
In this paper, we unify the Markov theory of a variety of different types of
graphs used in graphical Markov models by introducing the class of loopless
mixed graphs, and show that all independence models induced by -separation
on such graphs are compositional graphoids. We focus in particular on the
subclass of ribbonless graphs which as special cases include undirected graphs,
bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and
summary graphs. We define maximality of such graphs as well as a pairwise and a
global Markov property. We prove that the global and pairwise Markov properties
of a maximal ribbonless graph are equivalent for any independence model that is
a compositional graphoid.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ502 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Markov Properties of Discrete Determinantal Point Processes
Determinantal point processes (DPPs) are probabilistic models for repulsion.
When used to represent the occurrence of random subsets of a finite base set,
DPPs allow to model global negative associations in a mathematically elegant
and direct way. Discrete DPPs have become popular and computationally tractable
models for solving several machine learning tasks that require the selection of
diverse objects, and have been successfully applied in numerous real-life
problems. Despite their popularity, the statistical properties of such models
have not been adequately explored. In this note, we derive the Markov
properties of discrete DPPs and show how they can be expressed using graphical
models.Comment: 9 pages, 1 figur
On Finite Exchangeability and Conditional Independence
We study the independence structure of finitely exchangeable distributions
over random vectors and random networks. In particular, we provide necessary
and sufficient conditions for an exchangeable vector so that its elements are
completely independent or completely dependent. We also provide a sufficient
condition for an exchangeable vector so that its elements are marginally
independent. We then generalize these results and conditions for exchangeable
random networks. In this case, it is demonstrated that the situation is more
complex. We show that the independence structure of exchangeable random
networks lies in one of six regimes that are two-fold dual to one another,
represented by undirected and bidirected independence graphs in graphical model
sense with graphs that are complement of each other. In addition, under certain
additional assumptions, we provide necessary and sufficient conditions for the
exchangeable network distributions to be faithful to each of these graphs.Comment: 25 pages, 2 figure
Stable mixed graphs
In this paper, we study classes of graphs with three types of edges that
capture the modified independence structure of a directed acyclic graph (DAG)
after marginalisation over unobserved variables and conditioning on selection
variables using the -separation criterion. These include MC, summary, and
ancestral graphs. As a modification of MC graphs, we define the class of
ribbonless graphs (RGs) that permits the use of the -separation criterion.
RGs contain summary and ancestral graphs as subclasses, and each RG can be
generated by a DAG after marginalisation and conditioning. We derive simple
algorithms to generate RGs, from given DAGs or RGs, and also to generate
summary and ancestral graphs in a simple way by further extension of the
RG-generating algorithm. This enables us to develop a parallel theory on these
three classes and to study the relationships between them as well as the use of
each class.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ454 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Sequences of regressions and their independences
Ordered sequences of univariate or multivariate regressions provide
statistical models for analysing data from randomized, possibly sequential
interventions, from cohort or multi-wave panel studies, but also from
cross-sectional or retrospective studies. Conditional independences are
captured by what we name regression graphs, provided the generated distribution
shares some properties with a joint Gaussian distribution. Regression graphs
extend purely directed, acyclic graphs by two types of undirected graph, one
type for components of joint responses and the other for components of the
context vector variable. We review the special features and the history of
regression graphs, derive criteria to read all implied independences of a
regression graph and prove criteria for Markov equivalence that is to judge
whether two different graphs imply the same set of independence statements.
Knowledge of Markov equivalence provides alternative interpretations of a given
sequence of regressions, is essential for machine learning strategies and
permits to use the simple graphical criteria of regression graphs on graphs for
which the corresponding criteria are in general more complex. Under the known
conditions that a Markov equivalent directed acyclic graph exists for any given
regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited
discussion paper in the journal TES
On Exchangeability in Network Models
We derive representation theorems for exchangeable distributions on finite
and infinite graphs using elementary arguments based on geometric and
graph-theoretic concepts. Our results elucidate some of the key differences,
and their implications, between statistical network models that are finitely
exchangeable and models that define a consistent sequence of probability
distributions on graphs of increasing size.Comment: Dedicated to the memory of Steve Fienber
Axiomatization of Interventional Probability Distributions
Causal intervention is an essential tool in causal inference. It is
axiomatized under the rules of do-calculus in the case of structure causal
models. We provide simple axiomatizations for families of probability
distributions to be different types of interventional distributions. Our
axiomatizations neatly lead to a simple and clear theory of causality that has
several advantages: it does not need to make use of any modeling assumptions
such as those imposed by structural causal models; it only relies on
interventions on single variables; it includes most cases with latent variables
and causal cycles; and more importantly, it does not assume the existence of an
underlying true causal graph as we do not take it as the primitive object--in
fact, a causal graph is derived as a by-product of our theory. We show that,
under our axiomatizations, the intervened distributions are Markovian to the
defined intervened causal graphs, and an observed joint probability
distribution is Markovian to the obtained causal graph; these results are
consistent with the case of structural causal models, and as a result, the
existing theory of causal inference applies. We also show that a large class of
natural structural causal models satisfy the theory presented here. We note
that the aim of this paper is axiomatization of interventional families, which
is subtly different from "causal modeling."Comment: 39 pages, 4 figure