84 research outputs found
Learning Linear Non-Gaussian Polytree Models
In the context of graphical causal discovery, we adapt the versatile framework of linear non-Gaussian acyclic models (LiNGAMs) to propose new algorithms to efficiently learn graphs that are polytrees. Our approach combines the Chow--Liu algorithm, which first learns the undirected tree structure, with novel schemes to orient the edges. The orientation schemes assess algebraic relations among moments of the data-generating distribution and are computationally inexpensive. We establish high-dimensional consistency results for our approach and compare different algorithmic versions in numerical experiments
An efficiency upper bound for inverse covariance estimation
We derive an upper bound for the efficiency of estimating entries in the
inverse covariance matrix of a high dimensional distribution. We show that in
order to approximate an off-diagonal entry of the density matrix of a
-dimensional Gaussian random vector, one needs at least a number of samples
proportional to . Furthermore, we show that with samples, the
hypothesis that two given coordinates are fully correlated, when all other
coordinates are conditioned to be zero, cannot be told apart from the
hypothesis that the two are uncorrelated.Comment: 7 Page
A Mutagenetic Tree Hidden Markov Model for Longitudinal Clonal HIV Sequence Data
RNA viruses provide prominent examples of measurably evolving populations. In
HIV infection, the development of drug resistance is of particular interest,
because precise predictions of the outcome of this evolutionary process are a
prerequisite for the rational design of antiretroviral treatment protocols. We
present a mutagenetic tree hidden Markov model for the analysis of longitudinal
clonal sequence data. Using HIV mutation data from clinical trials, we estimate
the order and rate of occurrence of seven amino acid changes that are
associated with resistance to the reverse transcriptase inhibitor efavirenz.Comment: 20 pages, 6 figure
Generalized Permutohedra from Probabilistic Graphical Models
A graphical model encodes conditional independence relations via the Markov
properties. For an undirected graph these conditional independence relations
can be represented by a simple polytope known as the graph associahedron, which
can be constructed as a Minkowski sum of standard simplices. There is an
analogous polytope for conditional independence relations coming from a regular
Gaussian model, and it can be defined using multiinformation or relative
entropy. For directed acyclic graphical models and also for mixed graphical
models containing undirected, directed and bidirected edges, we give a
construction of this polytope, up to equivalence of normal fans, as a Minkowski
sum of matroid polytopes. Finally, we apply this geometric insight to construct
a new ordering-based search algorithm for causal inference via directed acyclic
graphical models.Comment: Appendix B is expanded. Final version to appear in SIAM J. Discrete
Mat
A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models
We propose a new method for the Maximum Likelihood Estimator (MLE) of
nonlinear mixed effects models when the variance matrix of Gaussian random
effects has a prescribed pattern of zeros (PPZ). The method consists in
coupling the recently developed Iterative Conditional Fitting (ICF) algorithm
with the Expectation Maximization (EM) algorithm. It provides positive definite
estimates for any sample size, and does not rely on any structural assumption
on the PPZ. It can be easily adapted to many versions of EM.Comment: Accepted for publication in Statistics and Computin
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
Separability problem for multipartite states of rank at most four
One of the most important problems in quantum information is the separability
problem, which asks whether a given quantum state is separable. We investigate
multipartite states of rank at most four which are PPT (i.e., all their partial
transposes are positive semidefinite). We show that any PPT state of rank two
or three is separable and has length at most four. For separable states of rank
four, we show that they have length at most six. It is six only for some
qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of
rank four is necessarily supported on a 3x3 or a 2x2x2 subsystem. We obtain a
very simple criterion for the separability problem of the PPT states of rank at
most four: such a state is entangled if and only if its range contains no
product vectors. This criterion can be easily applied since a four-dimensional
subspace in the 3x3 or 2x2x2 system contains a product vector if and only if
its Pluecker coordinates satisfy a homogeneous polynomial equation (the Chow
form of the corresponding Segre variety). We have computed an explicit
determinantal expression for the Chow form in the former case, while such
expression was already known in the latter case.Comment: 19 page
Dynamic Gaussian graphical models for modelling genomic networks
After sequencing the entire DNA for various organisms, the challenge has become understanding the functional interrelatedness of the genome. Only by understanding the pathways for various complex diseases can we begin to make sense of any type of treatment. Unfortu- nately, decyphering the genomic network structure is an enormous task. Even with a small number of genes the number of possible networks is very large. This problem becomes even more difficult, when we consider dynamical networks. We consider the problem of estimating a sparse dy- namic Gaussian graphical model with L1 penalized maximum likelihood of structured precision matrix. The structure can consist of specific time dynamics, known presence or absence of links in the graphical model or equality constraints on the parameters. The model is defined on the basis of partial correlations, which results in a specific class precision matrices. A priori L1 penalized maximum likelihood estimation in this class is extremely difficult, because of the above mentioned constraints, the computational complexity of the L1 constraint on the side of the usual positive-definite constraint. The implementation is non-trivial, but we show that the computation can be done effectively by taking advan- tage of an efficient maximum determinant algorithm developed in convex optimization
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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