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Algebraic factor analysis: tetrads, pentads and beyond
"... Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of po ..."
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Cited by 28 (12 self)
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Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. Gröbner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodnessoffit. 1
Binary models for marginal independence
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B
, 2005
"... A number of authors have considered multivariate Gaussian models for marginal independence. In this paper we develop models for binary data with the same independence structure. The models can be parameterized based on Möbius inversion and maximum likelihood estimation can be performed using a versi ..."
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Cited by 16 (2 self)
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A number of authors have considered multivariate Gaussian models for marginal independence. In this paper we develop models for binary data with the same independence structure. The models can be parameterized based on Möbius inversion and maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. The approach is illustrated on a simple example. Relations to multivariate logistic and dependence ratio models are discussed.
The Effect Restoration from Measurement Bias in Causal Inference
"... This paper highlights several areas where graphical techniques can be harnessed to address the problem of measurement errors in causal inference. In particulars,the paper discusses the control of partially observable confounders in parametric and non parametric models and the computational problem o ..."
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This paper highlights several areas where graphical techniques can be harnessed to address the problem of measurement errors in causal inference. In particulars,the paper discusses the control of partially observable confounders in parametric and non parametric models and the computational problem of obtaining biasfree effect estimates in such models.
Graphical Answers to . . .
, 2004
"... In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bidirected graphs are special instances of ancestral graphs, which are mixed ..."
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In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph (alternatively graphs with dashed edges have been used for this purpose). Bidirected graphs are special instances of ancestral graphs, which are mixed graphs with undirected, directed, and bidirected edges. In this paper, we show how simplicial sets and the newly defined orientable edges can be used to construct a maximal ancestral graph that is Markov equivalent to a given bidirected graph, i.e. the independence models associated with the two graphs coincide, and such that the number of arrowheads is minimal. Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. This construction yields an immediate check whether the original bidirected graph is Markov equivalent to a directed acyclic graph (Bayesian network) or an undirected graph (Markov random field). Moreover, the ancestral graph construction allows for computationally more efficient maximum likelihood fitting of covariance graph models, i.e. Gaussian bidirected graph models. In particular, we give a necessary and sufficient graphical criterion for determining when an entry of the maximum likelihood estimate of the covariance matrix must equal its empirical counterpart.
A MCMC Approach for Learning the Structure of Gaussian Acyclic Directed Mixed Graphs
"... Abstract Graphical models are widely used to encode conditional independence constraints and causal assumptions, the directed acyclic graph (DAG) being one of the most common types of models. However, DAGs are not closed under marginalization: that is, a chosen marginal of a distribution Markov to a ..."
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Abstract Graphical models are widely used to encode conditional independence constraints and causal assumptions, the directed acyclic graph (DAG) being one of the most common types of models. However, DAGs are not closed under marginalization: that is, a chosen marginal of a distribution Markov to a DAG might not be representable with another DAG, unless one discards some of the structural independencies. Acyclic directed mixed graphs (ADMGs) generalize DAGs so that closure under marginalization is possible. In a previous work, we showed how to perform Bayesian inference to infer the posterior distribution of the parameters of a given Gaussian ADMG model, where the graph is fixed. In this paper, we extend this procedure to allow for priors over graph structures.