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For general non-Gaussian distributions, the covariance and precision matrices do not encode the independence structure of the variables, as they do for the multivariate Gaussian. This paper builds on previous work to show that for a class…
This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis…
We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for…
We study the accuracy of estimating the covariance and the precision matrix of a $D$-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation…
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix…
The properties of the normal distribution under linear transformation, as well the easy way to compute the covariance matrix of marginals and conditionals, offer a unique opportunity to get an insight about several aspects of uncertainties…
The statistical distribution of levels of an integrable system is claimed to be a Poisson distribution. In this paper, we numerically generate an ensemble of N dimensional random diagonal matrices as a model for regular systems. We evaluate…
In a differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and…
Gaussian graphical models, where it is assumed that the variables of interest jointly follow a multivariate normal distribution with a sparse precision matrix, have been used to study intrinsic dependence among variables, but the normality…
A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent…
CoVariance Neural Networks (VNNs) perform convolutions on the graph determined by the covariance matrix of the data, which enables expressive and stable covariance-based learning. However, covariance matrices are typically dense, fail to…
Parametric distributions are an important part of statistics. There is now a voluminous literature on different fascinating formulations of flexible distributions. We present a selective and brief overview of a small subset of these…
A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge…
DAG models with hidden variables present many difficulties that are not present when all nodes are observed. In particular, fully observed DAG models are identified and correspond to well-defined sets ofdistributions, whereas this is not…
Nonparanormal models describe the joint distribution of multivariate responses via latent Gaussian, and thus parametric, copulae while allowing flexible nonparametric marginals. Some aspects of such distributions, for example conditional…
Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously…
In Gaussian graphical models, conditional independence and partial correlations are natural inferential targets for understanding direct relationships in multivariate data. No comparable framework exists for spatial processes, where…
We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…
Gaussian graphical models have been used to study intrinsic dependence among several variables, but the Gaussianity assumption may be restrictive in many applications. A nonparanormal graphical model is a semiparametric generalization for…