Harry Crane (Rutgers University)

Relative exchangeability

Symmetry arguments lie at the heart of classical considerations in inductive inference and statistics.  In statistics, de Finetti’s notion of exchangeability is the most prominent symmetry assumption, laying the foundation for Bayesian inference.  In practice, many statistical and scientific problems exhibit only partial symmetry determined by some underlying structure in a population.  As a simple example, consider measurements X_1, X_2, … taken on a population of men and Y_1, Y_2, … taken on a population of women.  Without further information, we may assume the distribution of (X_1,X_2,…; Y_1,Y_2,…) is symmetric under independent relabeling of the X’s and Y’s, but not under arbitrary relabeling of the entire sequence.  In general applications, the symmetries may be more complex, leading to the notion of `relative exchangeability’, a type of partial exchangeability which reflects distributional invariance with respect to the symmetries of another structure.  I will discuss recent work in this area, including a generic representation for relatively exchangeable structures and applications to the study of certain combinatorial stochastic processes, including coalescent and graph-valued processes.