Maximum Likelihood Estimate in Discrete Hierarchical Log-Linear Models

Hierarchical log-linear models are essential tools used for relationship identification between variables in complex high-dimensional problems. In this thesis we study two problems: the computation and the existence of the maximum likelihood estimate (henceforth abbreviated MLE) in high-dimensional hierarchical log-linear models.

When the number of variables is large, computing the MLE of the parameters is a difficult task to accomplish. A popular approach is to estimate the composite MLE rather than the MLE itself, that is, estimate the value of the parameter that maximizes the product of local conditional likelihoods. A more recent development is to choose the components of the composite likelihood to be local marginal likelihoods. We first show that the estimates obtained from local conditional and marginal likelihoods are identical. Second, we study the asymptotic properties of the composite MLE obtained by averaging the local estimates, under the double asymptotic regime, when both the dimension p and sample size N go to infinity. We compare the rate of convergence to the true parameter of the composite MLE with that of the global MLE under the same conditions. We also look at the asymptotic properties of the composite MLE when p is fixed and N goes to infinity and thus recover the same asymptotic results for p fixed as those given by Liu in 2012.

The existence of the MLE in hierarchical log-linear models has important consequences for statistical inference: estimation, confidence intervals and testing as we shall see. Determining whether this estimate exists is equivalent to finding whether the data belongs to the boundary of the marginal polytope of the model or not. In 2012, Fienberg and Rinaldo gave a linear programming method that determines the smallest such face for relatively low-dimensional models. In this thesis, we consider higher-dimensional problems. We develop the methology to obtain an outer and inner approximation to the smallest face of the marginal polytope containing the data vector. Outer approximations are obtained by looking at submodels of the original hierarchical model, and inner approximations are obtained by working with larger models.