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**1 - 4**of**4**### Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

"... Abstract We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models ..."

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Abstract We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models

### Inference for High-dimensional Exponential Family Graphical Models

"... Abstract Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on highdimensional estimation of exponential family graphical models, including Gaussian and Ising models, is focused on consistent model selection. However, the ..."

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Abstract Probabilistic graphical models have been widely used to model complex systems and aid scientific discoveries. Most existing work on highdimensional estimation of exponential family graphical models, including Gaussian and Ising models, is focused on consistent model selection. However, these results do not characterize uncertainty in the estimated structure and are of limited value to scientists who worry whether their findings will be reproducible and if the estimated edges are present in the model due to random chance. In this paper, we propose a novel estimator for edge parameters in an exponential family graphical models. We prove that the estimator is √ n-consistent and asymptotically Normal. This result allows us to construct confidence intervals for edge parameters, as well as, hypothesis tests. We establish our results under conditions that are typically assumed in the literature for consistent estimation. However, we do not require that the estimator consistently recovers the graph structure. In particular, we prove that the asymptotic distribution of the estimator is robust to model selection mistakes and uniformly valid for a large number of data-generating processes. We illustrate validity of our estimator through extensive simulation studies.

### Selection and Estimation for Mixed Graphical Models

, 2014

"... We consider the problem of estimating the parameters in a pairwise graphical model in which the distribution of each node, conditioned on the others, may have a different paramet-ric form. In particular, we assume that each node’s conditional distribution is in the exponential family. We identify re ..."

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We consider the problem of estimating the parameters in a pairwise graphical model in which the distribution of each node, conditioned on the others, may have a different paramet-ric form. In particular, we assume that each node’s conditional distribution is in the exponential family. We identify restrictions on the parameter space required for the existence of a well-defined joint density, and establish the consistency of the neighbourhood selection approach for graph reconstruction in high dimensions when the true underlying graph is sparse. Motivated by our theoretical results, we investigate the selection of edges between nodes whose condi-tional distributions take different parametric forms, and show that efficiency can be gained if edge estimates obtained from the regressions of particular nodes are used to reconstruct the graph. These results are illustrated with examples of Gaussian, Bernoulli, Poisson and expo-nential distributions. Our theoretical findings are corroborated by evidence from simulation studies.

### A General Framework for Mixed Graphical Models

"... “Mixed Data ” comprising a large number of heterogeneous variables (e.g. count, binary, continuous, skewed continuous, among other data types) are prevalent in varied areas such as genomics and proteomics, imaging genetics, national security, social networking, and Internet advertising. There have b ..."

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“Mixed Data ” comprising a large number of heterogeneous variables (e.g. count, binary, continuous, skewed continuous, among other data types) are prevalent in varied areas such as genomics and proteomics, imaging genetics, national security, social networking, and Internet advertising. There have been limited efforts at statistically modeling such mixed data jointly, in part because of the lack of computationally amenable multivariate distributions that can capture direct dependencies between such mixed variables of different types. In this paper, we address this by introducing a novel class of Block Directed Markov Random Fields (BDMRFs). Using the basic building block of node-conditional univariate exponential families from Yang et al. (2012), we introduce a class of mixed conditional random field distributions, that are then chained according to a block-directed acyclic graph to form our class of Block Directed Markov Random Fields (BDMRFs). The Markov independence graph structure underlying a BDMRF thus has both directed and undirected edges. We introduce conditions under which these distributions exist and are normalizable, study several instances of our models, and propose scalable penalized conditional likelihood estimators with statistical guarantees for recovering the underlying network structure. Simulations as well as an application to learning mixed genomic networks from next generation sequencing expression data and mutation data demonstrate the versatility of our methods.