Le jeudi 3 novembre 2011, 16:00, salle 1205 du Pavillon Burnside.

*Maximum likelihood estimation in network models*

Alessandro Rinaldo, Carnegie Mellon University

This talk is concerned with maximum likelihood estimation (MLE) in exponential statistical models for networks (random graphs) and, in particular, with the beta model, a simple model for undirected graphs in which the degree sequence is the minimal sufficient statistic. The speaker will present necessary and sufficient conditions for the existence of the MLE of the beta model parameters that are based on a geometric object known as the polytope of degree sequences. Using this result, it is possible to characterize in a combinatorial fashion sample points leading to a non-existent MLE and non-estimability of the probability parameters under a non-existent MLE. The speaker will further indicate some conditions guaranteeing that the MLE exists with probability tending to 1 as the number of nodes increases. Much of this analysis applies also to other well-known models for networks, such as the Rasch model, the Bradley-Terry model and the more general p1 model of Holland and Leinhardt. These results are in fact instantiations of rather general geometric properties of exponential families with polyhedral support that will be illustrated with a simple exponential random graph model.