14 research outputs found
Determining full conditional independence by low-order conditioning
A concentration graph associated with a random vector is an undirected graph
where each vertex corresponds to one random variable in the vector. The absence
of an edge between any pair of vertices (or variables) is equivalent to full
conditional independence between these two variables given all the other
variables. In the multivariate Gaussian case, the absence of an edge
corresponds to a zero coefficient in the precision matrix, which is the inverse
of the covariance matrix. It is well known that this concentration graph
represents some of the conditional independencies in the distribution of the
associated random vector. These conditional independencies correspond to the
"separations" or absence of edges in that graph. In this paper we assume that
there are no other independencies present in the probability distribution than
those represented by the graph. This property is called the perfect
Markovianity of the probability distribution with respect to the associated
concentration graph. We prove in this paper that this particular concentration
graph, the one associated with a perfect Markov distribution, can be determined
by only conditioning on a limited number of variables. We demonstrate that this
number is equal to the maximum size of the minimal separators in the
concentration graph.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ193 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A Localization Approach to Improve Iterative Proportional Scaling in Gaussian Graphical Models
We discuss an efficient implementation of the iterative proportional scaling
procedure in the multivariate Gaussian graphical models. We show that the
computational cost can be reduced by localization of the update procedure in
each iterative step by using the structure of a decomposable model obtained by
triangulation of the graph associated with the model. Some numerical
experiments demonstrate the competitive performance of the proposed algorithm.Comment: 12 page
On the causal interpretation of acyclic mixed graphs under multivariate normality
In multivariate statistics, acyclic mixed graphs with directed and bidirected
edges are widely used for compact representation of dependence structures that
can arise in the presence of hidden (i.e., latent or unobserved) variables.
Indeed, under multivariate normality, every mixed graph corresponds to a set of
covariance matrices that contains as a full-dimensional subset the covariance
matrices associated with a causally interpretable acyclic digraph. This digraph
generally has some of its nodes corresponding to hidden variables. We seek to
clarify for which mixed graphs there exists an acyclic digraph whose hidden
variable model coincides with the mixed graph model. Restricting to the
tractable setting of chain graphs and multivariate normality, we show that
decomposability of the bidirected part of the chain graph is necessary and
sufficient for equality between the mixed graph model and some hidden variable
model given by an acyclic digraph
A partial orthogonalization method for simulating covariance and concentration graph matrices
Structure learning methods for covariance and concentration graphs are often
validated on synthetic models, usually obtained by randomly generating: (i) an
undirected graph, and (ii) a compatible symmetric positive definite (SPD)
matrix. In order to ensure positive definiteness in (ii), a dominant diagonal
is usually imposed. However, the link strengths in the resulting graphical
model, determined by off-diagonal entries in the SPD matrix, are in many
scenarios extremely weak. Recovering the structure of the undirected graph thus
becomes a challenge, and algorithm validation is notably affected. In this
paper, we propose an alternative method which overcomes such problem yet
yielding a compatible SPD matrix. We generate a partially row-wise-orthogonal
matrix factor, where pairwise orthogonal rows correspond to missing edges in
the undirected graph. In numerical experiments ranging from moderately dense to
sparse scenarios, we obtain that, as the dimension increases, the link strength
we simulate is stable with respect to the structure sparsity. Importantly, we
show in a real validation setting how structure recovery is greatly improved
for all learning algorithms when using our proposed method, thereby producing a
more realistic comparison framework.Comment: 12 pages, 5 figures, conferenc
The Maximum Likelihood Threshold of a Path Diagram
Linear structural equation models postulate noisy linear relationships
between variables of interest. Each model corresponds to a path diagram, which
is a mixed graph with directed edges that encode the domains of the linear
functions and bidirected edges that indicate possible correlations among noise
terms. Using this graphical representation, we determine the maximum likelihood
threshold, that is, the minimum sample size at which the likelihood function of
a Gaussian structural equation model is almost surely bounded. Our result
allows the model to have feedback loops and is based on decomposing the path
diagram with respect to the connected components of its bidirected part. We
also prove that if the sample size is below the threshold, then the likelihood
function is almost surely unbounded. Our work clarifies, in particular, that
standard likelihood inference is applicable to sparse high-dimensional models
even if they feature feedback loops
Sparse Matrix Decompositions and Graph Characterizations
The question of when zeros (i.e., sparsity) in a positive definite matrix
are preserved in its Cholesky decomposition, and vice versa, was addressed by
Paulsen et al. in the Journal of Functional Analysis (85, pp151-178). In
particular, they prove that for the pattern of zeros in to be retained in
the Cholesky decomposition of , the pattern of zeros in has to
necessarily correspond to a chordal (or decomposable) graph associated with a
specific type of vertex ordering. This result therefore yields a
characterization of chordal graphs in terms of sparse positive definite
matrices. It has also proved to be extremely useful in probabilistic and
statistical analysis of Markov random fields where zeros in positive definite
correlation matrices are intimately related to the notion of stochastic
independence. Now, consider a positive definite matrix and its Cholesky
decomposition given by , where is lower triangular with unit
diagonal entries, and a diagonal matrix with positive entries. In this
paper, we prove that a necessary and sufficient condition for zeros (i.e.,
sparsity) in a positive definite matrix to be preserved in its associated
Cholesky matrix , \, and in addition also preserved in the inverse of the
Cholesky matrix , is that the pattern of zeros corresponds to a
co-chordal or homogeneous graph associated with a specific type of vertex
ordering. We proceed to provide a second characterization of this class of
graphs in terms of determinants of submatrices that correspond to cliques in
the graph. These results add to the growing body of literature in the field of
sparse matrix decompositions, and also prove to be critical ingredients in the
probabilistic analysis of an important class of Markov random fields
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