The Numeraire e-variable and Reverse Information Projection
AADITYA RAMDAS – CARNEGIE MELLON UNIVERSITY
ABSTRACT
In an excellent 1999 Yale PhD thesis, Jonathan Li proposed and defined a critical concept that he called the reverse information projection (RIPr), which is akin to a Kullback-Leibler projection of a distribution onto a set of probability measures. This concept has gained prominence recently in game-theoretic statistics and sequential testing by betting, because it characterizes the log-optimal bet/e-variable of a point alternative hypothesis against a composite null hypothesis. However, it required assumptions of convexity of the set of distributions and a common reference measure to define densities. In this talk, we will show how to fully and completely generalize the theory underlying the RIPr, showing that it is always well defined, without *any* assumptions. Further, a strong duality result identifies it as the dual to an optimal bet/e-variable called the numeraire, which is unique and also always exists without assumptions. This fully generalizes classical Kelly betting to composite nulls.
The talk will not assume any prior knowledge on these topics. This is joint work with Martin Larsson and Johannes Ruf.