The Stability Properties Of Countable Discrete Exponential Distribution Family

Now exponential random graph models are widely used to analyze network data.Exponential random graph models are exponential families,and so they inherit all the familiar virtues of exponential families in general.They are analytically and inferentially convenient.In terms of statistical computing,the most important obstacle is the fact that relational data tend to be dependent and discrete exponential families for dependent data come with intractable likelihood functions.As Markov chain Monte Carlo(MCMC)is the foremost means to generate draws from distributions with support Y,MCMC is key to both simulation and statistical inference.In applications to dependent network 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.Schweinberger(2011)introduced the notion of instability with finite discrete weighted exponential family models and suffi-cient statistics.Meanwhile they show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy.In special cases,the sub-set 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.The main content of this paper is to extend the notion of instability with finite discrete weighted exponential family models to the case of countable weighted edges,and to obtain the notion of instability with countable discrete exponential family and sufficient statistics.And show that unstable countable discrete exponential family mod-els are characterized by excessive sensitivity and near-degeneracy.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.