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Register a new userAutomorphism groups of Gaussian chain graph models
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Added by
Piotr Zwiernik
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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In this paper we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs. We discuss the maximal group which leaves a given model invariant and provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants and robustness. The computation of the group requires finding the essential graph. However, by applying Studenys theory of imsets we show that computations for DAGs can be performed efficiently without building the essential graph. In our proof we derive simple necessary and sufficient conditions on vanishing sub-minors of the concentration matrix in the model.
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We study a problem of estimation of smooth functionals of parameter $theta $ of Gaussian shift model $$ X=theta +xi, theta in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $theta$ in Gaussian noise $xi$ with zero mean and known covariance operator $Sigma.$ In particular, we develop estimators $T(X)$ of $f(theta)$ for functionals $f:Emapsto {mathbb R}$ of Holder smoothness $s>0$ such that $$ sup_{|theta|leq 1} {mathbb E}_{theta}(T(X)-f(theta))^2 lesssim Bigl(|Sigma| vee ({mathbb E}|xi|^2)^sBigr)wedge 1, $$ where $|Sigma|$ is the operator norm of $Sigma,$ and show that this mean squared error rate is minimax optimal at least in the case of standard Gaussian shift model ($E={mathbb R}^d$ equipped with the canonical Euclidean norm, $xi =sigma Z,$ $Zsim {mathcal N}(0;I_d)$). Moreover, we determine a sharp threshold on the smoothness $s$ of functional $f$ such that, for all $s$ above the threshold, $f(theta)$ can be estimated efficiently with a mean squared error rate of the order $|Sigma|$ in a small noise setting (that is, when ${mathbb E}|xi|^2$ is small). The construction of efficient estimators is crucially based on a bootstrap chain method of bias reduction. The results could be applied to a variety of special high-dimensional and infinite-dimensional Gaussian models (for vector, matrix and functional data).
The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focusing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGMs expressive power. These results are actually special cases of more general results about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses.
Let $(Y,(X_i)_{iinmathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of $(X_i)_{iin mathcal{I}backslash V}$ conditionally to $(X_i)_{iin V}$ against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of $X$ or the variance of $Y$ and applies in a high-dimensional setting. It straightforwardly extends to test the neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic properties of the test and we prove that it is rate optimal (up to a possible $log(n)$ factor) over various classes of alternatives under some additional assumptions. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.
We study parameter identifiability of directed Gaussian graphical models with one latent variable. In the scenario we consider, the latent variable is a confounder that forms a source node of the graph and is a parent to all other nodes, which correspond to the observed variables. We give a graphical condition that is sufficient for the Jacobian matrix of the parametrization map to be full rank, which entails that the parametrization is generically finite-to-one, a fact that is sometimes also referred to as local identifiability. We also derive a graphical condition that is necessary for such identifiability. Finally, we give a condition under which generic parameter identifiability can be determined from identifiability of a model associated with a subgraph. The power of these criteria is assessed via an exhaustive algebraic computational study on models with 4, 5, and 6 observable variables.
Let $X^{(n)}$ be an observation sampled from a distribution $P_{theta}^{(n)}$ with an unknown parameter $theta,$ $theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(theta)$ for a functional $f:Emapsto {mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}sim P_{theta}^{(n)}.$ Assuming that there exists an estimator $hat theta_n=hat theta_n(X^{(n)})$ of parameter $theta$ such that $sqrt{n}(hat theta_n-theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:Emapsto {mathbb R}$ such that $g(hat theta_n)$ is an asymptotically normal estimator of $f(theta)$ with $sqrt{n}$ rate provided that $s>frac{1}{1-alpha}$ and $dleq n^{alpha}$ for some $alphain (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(hat theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $sqrt{n}(hat theta_n-theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
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