569 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
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
Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property
The AMP Markov property is a recently proposed alternative Markov property
for chain graphs. In the case of continuous variables with a joint multivariate
Gaussian distribution, it is the AMP rather than the earlier introduced LWF
Markov property that is coherent with data-generation by natural
block-recursive regressions. In this paper, we show that maximum likelihood
estimates in Gaussian AMP chain graph models can be obtained by combining
generalized least squares and iterative proportional fitting to an iterative
algorithm. In an appendix, we give useful convergence results for iterative
partial maximization algorithms that apply in particular to the described
algorithm.Comment: 15 pages, article will appear in Scandinavian Journal of Statistic
Estimation of a Covariance Matrix with Zeros
We consider estimation of the covariance matrix of a multivariate random
vector under the constraint that certain covariances are zero. We first present
an algorithm, which we call Iterative Conditional Fitting, for computing the
maximum likelihood estimator of the constrained covariance matrix, under the
assumption of multivariate normality. In contrast to previous approaches, this
algorithm has guaranteed convergence properties. Dropping the assumption of
multivariate normality, we show how to estimate the covariance matrix in an
empirical likelihood approach. These approaches are then compared via
simulation and on an example of gene expression.Comment: 25 page
Distributional Equivalence and Structure Learning for Bow-free Acyclic Path Diagrams
We consider the problem of structure learning for bow-free acyclic path
diagrams (BAPs). BAPs can be viewed as a generalization of linear Gaussian DAG
models that allow for certain hidden variables. We present a first method for
this problem using a greedy score-based search algorithm. We also prove some
necessary and some sufficient conditions for distributional equivalence of BAPs
which are used in an algorithmic ap- proach to compute (nearly) equivalent
model structures. This allows us to infer lower bounds of causal effects. We
also present applications to real and simulated datasets using our publicly
available R-package
Discrete chain graph models
The statistical literature discusses different types of Markov properties for
chain graphs that lead to four possible classes of chain graph Markov models.
The different models are rather well understood when the observations are
continuous and multivariate normal, and it is also known that one model class,
referred to as models of LWF (Lauritzen--Wermuth--Frydenberg) or block
concentration type, yields discrete models for categorical data that are
smooth. This paper considers the structural properties of the discrete models
based on the three alternative Markov properties. It is shown by example that
two of the alternative Markov properties can lead to non-smooth models. The
remaining model class, which can be viewed as a discrete version of
multivariate regressions, is proven to comprise only smooth models. The proof
employs a simple change of coordinates that also reveals that the model's
likelihood function is unimodal if the chain components of the graph are
complete sets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ172 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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