False vs. Missed Discoveries, Gaussian Decision Theory, and the Donsker-Varadhan Principle
The article touches on two themes in which Larry Brown has been interested, namely foundations, and mathematical analysis of Bayesian deci- sion theory. In the section on foundations, a new formulation of the problem of testing is given, and is theoretically explored. The formulation strikes a balance between false discoveries and missed discoveries. Basic higher order asymptotic theory for the α level that this formulation would imply is then worked out. Accuracy is investigated and examples are given. In the section on Bayesian decision theory, first, the Brown identities are connected to a set of inequalities of elliptic boundary value problems. It is shown by four specific results that sometimes a result in that field can lead to new results in Bayesian decision theory, and sometimes a result in decision theory can give surprisingly useful information about a problem in that field. For instance, Brown identities can provide amazingly good estimates of best constants in the Nash inequalities over Sobolev spaces. The article ends with a result on the modern theory of high dimensional Gaussian mean estimation. By means of a triangulation of the Brown identity, the Rayleigh-Ritz variational formula of boundary value problems, and the famous Donsker-Varadhan result connecting the Rayleigh-Ritz formula to the absorption time of a Brownian motion into the boundary of a smooth bounded open domain, we show that the minimax risk of estimating the Gaussian mean can be approximated by chasing a Brownian motion to the boundary of the parameter space. This link should be tested by simulation. 1. Foreword Although my failing eyesight has limited my normal activities, when the editors of this volume kindly invited me to write an article, I knew that it was an honor that I could not pass. I first corresponded with Larry Brown around 1982, when I was a PhD student at the Indian Statistical Institute in Calcutta. In the 30 years that have followed, I have learned from and have been inspired by Larry Brown's signature humility, and his work in numerous areas, including decision theory and mathematical analysis of difficult Bayesian problems, foundations, asymptotics, and theoretical validation of methodologies. This article touches on two of these themes, which form an extremely small corner of Larry Brown's work. These are among the main topics on which I have myself corresponded with him, or have worked with him. It is my greatest pleasure to present these few calculations in honor of Professor Brown.