Instability, Sensitivity, and Degeneracy of Discrete Exponential Families

In applications to dependent data, first and foremost relational data, a number of discrete exponential family models has turned out to be near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We introduce the notion of instability with an eye to characterize, detect, and penalize discrete exponential family models that are near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter space corresponding to non-degenerate distributions and mean-value parameters far from the boundary of the mean-value parameter space turns out to be a lower-dimensional subspace of the natural parameter space. These characteristics of unstable discrete exponential family models tend to obstruct Markov chain Monte Carlo simulation and statistical inference. In applications to relational data, we show that discrete exponential family models with Markov dependence tend to be unstable and that the parameter space of some curved exponential families contains unstable subsets.