Total positivity in structured binary distributions

05/01/2019
by   Steffen Lauritzen, et al.
0

We study binary distributions that are multivariate totally positive of order 2 (MTP2). Binary distributions can be represented as an exponential family and we show that MTP2 exponential families are convex. Moreover, MTP2 quadratic exponential families, which contain ferromagnetic Ising models and attractive Gaussian graphical models, 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 the sign patterns (1,-1) and (-1,1) are represented in the sample for every pair of vertices; in particular, this implies that the MLE may exist with n=d samples, in stark contrast to unrestricted binary exponential families where 2^d samples are required. Finally, we provide a 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. Throughout, we compare our results on MTP2 Ising models with the Gaussian case and identify similarities and differences.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/30/2013

Graphical Models and Exponential Families

We provide a classification of graphical models according to their repre...
research
02/18/2021

Learning Continuous Exponential Families Beyond Gaussian

We address the problem of learning of continuous exponential family dist...
research
11/28/2021

An inverse Sanov theorem for curved exponential families

We prove the large deviation principle (LDP) for posterior distributions...
research
11/15/2017

Kernel Conditional Exponential Family

A nonparametric family of conditional distributions is introduced, which...
research
05/19/2015

Vector-Space Markov Random Fields via Exponential Families

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of...
research
10/29/2020

Staged trees are curved exponential families

Staged tree models are a discrete generalization of Bayesian networks. W...
research
08/04/2021

Sparse Continuous Distributions and Fenchel-Young Losses

Exponential families are widely used in machine learning; they include m...

Please sign up or login with your details

Forgot password? Click here to reset