Results 1  10
of
19
Discrete chain graph models
 Bernoulli
, 2009
"... 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 mode ..."
Abstract

Cited by 37 (2 self)
 Add to MetaCart
(Show Context)
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 nonsmooth 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.
Likelihood ratio tests and singularities
 Ann. Statist
, 2008
"... Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisf ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
(Show Context)
Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semialgebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff’s theorem hold for semialgebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point. At boundary points or singularities, the tangent cone need not be a linear space and limiting distributions other than chisquare distributions may arise. While boundary points often lead to mixtures of chisquare distributions, singularities give rise to nonstandard limits. We demonstrate that minima of chisquare random variables are important for locally identifiable models, and in a study of the factor analysis model with one factor, we reveal connections to eigenvalues of Wishart matrices.
Graphical methods for efficient likelihood inference in gaussian covariance models
 Journal of Machine Learning
, 2008
"... Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the origi ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the original bidirected graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts. 1.
Symmetric measures via moments
, 2006
"... Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a fini ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (nonsingular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and loglinear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.
Tropical Implicitization
, 2010
"... In recent years, tropical geometry has developed as a theory on its own. Its two main aims are to answer open questions in algebraic geometry and to give new proofs of celebrated classical results. The main subject of this thesis is concerned with the former: the solution of implicitization problems ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
In recent years, tropical geometry has developed as a theory on its own. Its two main aims are to answer open questions in algebraic geometry and to give new proofs of celebrated classical results. The main subject of this thesis is concerned with the former: the solution of implicitization problems via tropical geometry. We develop new and explicit techniques that completely solve these challenges in four concrete examples. We start by studying a family of challenging examples inspired by algebraic statistics and machine learning: the restricted Boltzmann machines F(n, k). These machines are highly structured projective varieties in tensor spaces. They correspond to a statistical model encoded by the complete bipartite graph Kk,n, by marginalizing k of the n+ k binary random variables. In Chapter 2, we investigate this problem in the most general setting. We conjecture a formula for the expected dimension of the model, verifying it in all relevant cases. We also study inference functions and their interplay with tropicalization of polynomial maps. In Chapter 3, we focus on the particular case F(4, 2), answering a question by Drton, Sturmfels and Sullivant regarding the degree (and Newton polytope) of the homogeneous equation in 16 variables defining this model. We show that its degree is 110 and compute
The geometry of conditional independence tree models with hidden variables
 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY, CA 94720, USA EMAIL ADDRESS: MACUETO@MATH.BERKELEY.EDU DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY
, 2009
"... In this paper we investigate the geometry of undirected graphical models of trees when all the variables in the system are binary and some of them are hidden. We obtain a full description of those models which is given by polynomial equations and inequalities and give exact formulas for their param ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In this paper we investigate the geometry of undirected graphical models of trees when all the variables in the system are binary and some of them are hidden. We obtain a full description of those models which is given by polynomial equations and inequalities and give exact formulas for their parameters in terms of the marginal probability over the observed variables. We also show how correlations link to tree metrics considered in phylogenetics. Finally, a new system of coordinates is given that is intrinsically related to the phylogenetic tree models and which allows us to classify phylogenetic invariants.
Gaussian tree constraints applied to acoustic linguistic functional data
"... Evolutionary models of languages are usually considered to take the form of trees. With the development of socalled tree constraints the plausibility of the tree model assumptions can be addressed by checking whether the moments of observed variables lie within regions consistent with trees. In ou ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Evolutionary models of languages are usually considered to take the form of trees. With the development of socalled tree constraints the plausibility of the tree model assumptions can be addressed by checking whether the moments of observed variables lie within regions consistent with trees. In our linguistic application, the data set comprises acoustic samples (audio recordings) from speakers of five Romance languages or dialects. We wish to assess these functional data for compatibility with a hereditary tree model at the language level. A novel combination of canonical function analysis (CFA) with a separable covariance structure provides a method for generating a representative basis for the data. This resulting basis is formed of components which emphasize language differences whilst maintaining the integrity of the observational languagegroupings. A previously unexploited Gaussian tree constraint is then applied to componentbycomponent projections of the data to investigate adherence to an evolutionary tree. The results indicate that while a tree model is unlikely to be suitable for modeling all aspects of the acoustic linguistic data, certain features of the spoken Romance languages highlighted by the separableCFA basis may indeed be suitably modeled as a tree.