Since Wald's seminal work on statistical decision theory in the 1940s and 1950s, statisticians have been trying to relate Bayesian and frequentist notions of optimality. That work---carried out by Wald, LeCam, Blackwell, Stein, Berger, Brown, and many others---never succeeded in equating Bayesian and frequentist notions without strong technical and regularity conditions. For example, every admissible procedure is Bayes on a finite parameter space, but this is no longer the case once you move to noncompact spaces. 

In this session, we discuss recent work that succeeds in exact unification for notions of admissibility by allowing Bayesian priors that assign infinitesimal probability to sets. This work, formalized using Nonstandard Analysis, raises the possibility of a broad unification. 

After a short introduction by Daniel Roy, we will get a brief crash course on nonstandard analysis by David Schrittesser, followed by a description of recent work unifying Frequentist admissibility and Bayes optimality. 

Next, Aaron Smith will describe work on the existence of exact matching priors (producing Bayesian credible sets with confidence coverage) on compact spaces, proven using nonstandard analysis.

Finally, Haosui Duanmu will discuss work on misspecification and the dynamics of Bayesian updating, showing the existence of equilibriums under broader conditions, again using nonstandard analysis.

Session Chair: Daniel Roy