Exponential Families of Probability Measures

09 Jan 2024 13:30

I should explain what these area, but having done so elsewhere, I am feeling disinclined to do it again. (Later, I should just copy that text.)

I am particularly interested in exponential families for time series (very natural for Markov models) and for networks. More generally, if I have a family of stochastic processes (collections of dependent random variables) which form exponential families, what constraints does that put on the process?

Exponential families correspond to canonical ensembles in statistical mechanics. (Natural sufficient statistics : natural parameters :: extensive macroscopic variables : conjugate intensive variables.) In statistical mechanics, one of the justifications for using canonical ensembles for large systems comes from large deviations theory. Is there something equivalent in statistics proper? (Roussas's results on local asymptotic approximation of parametric models by exponential families feels like it should be connected here.)