859 research outputs found
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
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
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
The Grow-Shrink strategy for learning Markov network structures constrained by context-specific independences
Markov networks are models for compactly representing complex probability
distributions. They are composed by a structure and a set of numerical weights.
The structure qualitatively describes independences in the distribution, which
can be exploited to factorize the distribution into a set of compact functions.
A key application for learning structures from data is to automatically
discover knowledge. In practice, structure learning algorithms focused on
"knowledge discovery" present a limitation: they use a coarse-grained
representation of the structure. As a result, this representation cannot
describe context-specific independences. Very recently, an algorithm called
CSPC was designed to overcome this limitation, but it has a high computational
complexity. This work tries to mitigate this downside presenting CSGS, an
algorithm that uses the Grow-Shrink strategy for reducing unnecessary
computations. On an empirical evaluation, the structures learned by CSGS
achieve competitive accuracies and lower computational complexity with respect
to those obtained by CSPC.Comment: 12 pages, and 8 figures. This works was presented in IBERAMIA 201
Concepts and a case study for a flexible class of graphical Markov models
With graphical Markov models, one can investigate complex dependences,
summarize some results of statistical analyses with graphs and use these graphs
to understand implications of well-fitting models. The models have a rich
history and form an area that has been intensively studied and developed in
recent years. We give a brief review of the main concepts and describe in more
detail a flexible subclass of models, called traceable regressions. These are
sequences of joint response regressions for which regression graphs permit one
to trace and thereby understand pathways of dependence. We use these methods to
reanalyze and interpret data from a prospective study of child development, now
known as the Mannheim Study of Children at Risk. The two related primary
features concern cognitive and motor development, at the age of 4.5 and 8 years
of a child. Deficits in these features form a sequence of joint responses.
Several possible risks are assessed at birth of the child and when the child
reached age 3 months and 2 years.Comment: 21 pages, 7 figures, 7 tables; invited, refereed chapter in a boo
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