Rao-Blackwellized e-variables

TL;DR

This paper demonstrates that conditioning on sufficient statistics can only increase the expected utility of e-variables, extending the Rao-Blackwell theorem to this context and providing practical applications like simplified derivations in linear regression.

Contribution

It establishes a Rao-Blackwell type theorem for e-variables, showing utility improvement after conditioning, with broad implications and applications.

Findings

Expected utility of e-variables increases after conditioning on sufficient statistics.

The result applies to compound, asymptotic e-variables, and e-processes.

Simplifies derivation of log-optimal e-variables in linear regression.

Abstract

We show that for any concave utility, the expected utility of an e-variable can only increase after conditioning on a sufficient statistic. The simplest form of the result has an extremely straightforward proof, which follows from a single application of Jensen's inequality. Similar statements hold for compound e-variables, asymptotic e-variables, and e-processes. These results echo the Rao-Blackwell theorem, which states that the expected squared error of an estimator can only decrease after conditioning on a sufficient statistic. We provide several applications of this insight, including a simplified derivation of the log-optimal e-variable for linear regression with known variance.