On Graphical Models via Univariate Exponential Family Distributions

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential families. This allows us to derive multivariate graphical model distributions from univariate exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributions.Comment: Journal of Machine Learning Researc

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if both of the sign patterns $(1,-1)$ and $(-1,1)$ are represented in the sample for every pair of variables; in particular, this implies that the MLE may exist with $n=d$ observations, in stark contrast to unrestricted binary exponential families where $2^d$ observations are required. Finally, we provide a novel and globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders

Linear Estimating Equations for Exponential Families with Application to Gaussian Linear Concentration Models

In many families of distributions, maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available. The score matching estimator of Hyv\"arinen (2005) provides an alternative where this normalization constant is not required. The corresponding estimating equations become linear for an exponential family. The score matching estimator is shown to be consistent and asymptotically normally distributed for such models, although not necessarily efficient. Gaussian linear concentration models are examples of such families. For linear concentration models that are also linear in the covariance we show that the score matching estimator is identical to the maximum likelihood estimator, hence in such cases it is also efficient. Gaussian graphical models and graphical models with symmetries form particularly interesting subclasses of linear concentration models and we investigate the potential use of the score matching estimator for this case

Algebraic Geometry of Quantum Graphical Models

Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.Comment: 20 pages, comments welcome

Stratified Gaussian graphical models

Gaussian graphical models represent the backbone of the statistical toolbox for analyzing continuous multivariate systems. However, due to the intrinsic properties of the multivariate normal distribution, use of this model family may hide certain forms of context-specific independence that are natural to consider from an applied perspective. Such independencies have been earlier introduced to generalize discrete graphical models and Bayesian networks into more flexible model families. Here, we adapt the idea of context-specific independence to Gaussian graphical models by introducing a stratification of the Euclidean space such that a conditional independence may hold in certain segments but be absent elsewhere. It is shown that the stratified models define a curved exponential family, which retains considerable tractability for parameter estimation and model selection.Peer reviewe

Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

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 of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1162 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org