The long-term goal of this project has been the development of useful tools for scientific inference, both in probability and statistics. The present proposal focuses on exponential families of probability distributions, especially as they apply to complex applications such as graphical models and high-dimensional Bayesian/frequentist computation. Specific projects include exponential families for graphs, including the effects of covariate information, and ways of using the frequentist bootstrap to compute Bayesian inferential distributions.
Exponential families include most of the well-known probability distributions -- binomial, Poisson, normal etc, -- so exponential family results have a wide scope of application. They live at the core of applied and theoretical statistics, and are of growing importance in probability theory. Graphical models, which include studies of the internet, gene expression data, and social networks such as Facebook, use exponential family structure to simplify analysis of the enormous data sets involved. Bayesian applications of the theory, increasingly popular in the "Big Data" world, are investigated here in terms of the bootstrap, a frequentist device, with the goal of connecting the two approaches.
