2,468 research outputs found
Mixed Cumulative Distribution Networks
Directed acyclic graphs (DAGs) are a popular framework to express
multivariate probability distributions. Acyclic directed mixed graphs (ADMGs)
are generalizations of DAGs that can succinctly capture much richer sets of
conditional independencies, and are especially useful in modeling the effects
of latent variables implicitly. Unfortunately there are currently no good
parameterizations of general ADMGs. In this paper, we apply recent work on
cumulative distribution networks and copulas to propose one one general
construction for ADMG models. We consider a simple parameter estimation
approach, and report some encouraging experimental results.Comment: 11 pages, 4 figure
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
Graphical Markov models: overview
We describe how graphical Markov models started to emerge in the last 40
years, based on three essential concepts that had been developed independently
more than a century ago. Sequences of joint or single regressions and their
regression graphs are singled out as being best suited for analyzing
longitudinal data and for tracing developmental pathways. Interpretations are
illustrated using two sets of data and some of the more recent, important
results for sequences of regressions are summarized.Comment: 22 pages, 9 figure
Marginal log-linear parameters for graphical Markov models
Marginal log-linear (MLL) models provide a flexible approach to multivariate
discrete data. MLL parametrizations under linear constraints induce a wide
variety of models, including models defined by conditional independences. We
introduce a sub-class of MLL models which correspond to Acyclic Directed Mixed
Graphs (ADMGs) under the usual global Markov property. We characterize for
precisely which graphs the resulting parametrization is variation independent.
The MLL approach provides the first description of ADMG models in terms of a
minimal list of constraints. The parametrization is also easily adapted to
sparse modelling techniques, which we illustrate using several examples of real
data.Comment: 36 page
Graphical Markov models, unifying results and their interpretation
Graphical Markov models combine conditional independence constraints with
graphical representations of stepwise data generating processes.The models
started to be formulated about 40 years ago and vigorous development is
ongoing. Longitudinal observational studies as well as intervention studies are
best modeled via a subclass called regression graph models and, especially
traceable regressions. Regression graphs include two types of undirected graph
and directed acyclic graphs in ordered sequences of joint responses. Response
components may correspond to discrete or continuous random variables and may
depend exclusively on variables which have been generated earlier. These
aspects are essential when causal hypothesis are the motivation for the
planning of empirical studies.
To turn the graphs into useful tools for tracing developmental pathways and
for predicting structure in alternative models, the generated distributions
have to mimic some properties of joint Gaussian distributions. Here, relevant
results concerning these aspects are spelled out and illustrated by examples.
With regression graph models, it becomes feasible, for the first time, to
derive structural effects of (1) ignoring some of the variables, of (2)
selecting subpopulations via fixed levels of some other variables or of (3)
changing the order in which the variables might get generated. Thus, the most
important future applications of these models will aim at the best possible
integration of knowledge from related studies.Comment: 34 Pages, 11 figures, 1 tabl
Smooth, identifiable supermodels of discrete DAG models with latent variables
We provide a parameterization of the discrete nested Markov model, which is a
supermodel that approximates DAG models (Bayesian network models) with latent
variables. Such models are widely used in causal inference and machine
learning. We explicitly evaluate their dimension, show that they are curved
exponential families of distributions, and fit them to data. The
parameterization avoids the irregularities and unidentifiability of latent
variable models. The parameters used are all fully identifiable and
causally-interpretable quantities.Comment: 30 page
Margins of discrete Bayesian networks
Bayesian network models with latent variables are widely used in statistics
and machine learning. In this paper we provide a complete algebraic
characterization of Bayesian network models with latent variables when the
observed variables are discrete and no assumption is made about the state-space
of the latent variables. We show that it is algebraically equivalent to the
so-called nested Markov model, meaning that the two are the same up to
inequality constraints on the joint probabilities. In particular these two
models have the same dimension. The nested Markov model is therefore the best
possible description of the latent variable model that avoids consideration of
inequalities, which are extremely complicated in general. A consequence of this
is that the constraint finding algorithm of Tian and Pearl (UAI 2002,
pp519-527) is complete for finding equality constraints.
Latent variable models suffer from difficulties of unidentifiable parameters
and non-regular asymptotics; in contrast the nested Markov model is fully
identifiable, represents a curved exponential family of known dimension, and
can easily be fitted using an explicit parameterization.Comment: 41 page
Binary Models for Marginal Independence
Log-linear models are a classical tool for the analysis of contingency
tables. In particular, the subclass of graphical log-linear models provides a
general framework for modelling conditional independences. However, with the
exception of special structures, marginal independence hypotheses cannot be
accommodated by these traditional models. Focusing on binary variables, we
present a model class that provides a framework for modelling marginal
independences in contingency tables. The approach taken is graphical and draws
on analogies to multivariate Gaussian models for marginal independence. For the
graphical model representation we use bi-directed graphs, which are in the
tradition of path diagrams. We show how the models can be parameterized in a
simple fashion, and how maximum likelihood estimation can be performed using a
version of the Iterated Conditional Fitting algorithm. Finally we consider
combining these models with symmetry restrictions
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