A local approach to estimation in discrete loglinear models

TL;DR

This paper introduces a local approach to identify faces of the marginal cone for high-dimensional discrete graphical models and analyzes the properties of composite maximum likelihood estimates in large-scale settings.

Contribution

It presents a method to identify marginal cone faces via local subgraph models and compares local likelihood approaches, including asymptotic analysis of composite likelihood estimates.

Findings

A face of the marginal cone can be identified using local subgraph models.

Consensus between local conditional and marginal likelihood estimates is established.

Asymptotic properties of composite likelihood estimates are analyzed for large models and samples.

Abstract

We consider two connected aspects of maximum likelihood estimation of the parameter for high-dimensional discrete graphical models: the existence of the maximum likelihood estimate (mle) and its computation. When the data is sparse, there are many zeros in the contingency table and the maximum likelihood estimate of the parameter may not exist. Fienberg and Rinaldo (2012) have shown that the mle does not exists iff the data vector belongs to a face of the so-called marginal cone spanned by the rows of the design matrix of the model. Identifying these faces in high-dimension is challenging. In this paper, we take a local approach : we show that one such face, albeit possibly not the smallest one, can be identified by looking at a collection of marginal graphical models generated by induced subgraphs $G_i,i=1,\ldots,k$ of $G$. This is our first contribution. Our second contribution concerns the composite maximum likelihood estimate. When the dimension of the problem is large, estimating the parameters of a given graphical model through maximum likelihood is challenging, if not impossible. The traditional approach to this problem has been local with the use of composite likelihood based on local conditional likelihoods. A more recent development is to have the components of the composite likelihood be marginal likelihoods centred around each $v$. We first show that the estimates obtained by consensus through local conditional and marginal likelihoods are identical. We then study the asymptotic properties of the composite maximum likelihood estimate when both the dimension of the model and the sample size $N$ go to infinity.