GRO e-variable

GRO stands for “growth rate optimality”. The term was coined by Grunwald, De Heide, and Koolen (2023).

Consider a simple alternative. This says to find the e-value which maximizes ${\mathbb{E}}_Q[\log E_{\tau }]$ under the alternative $Q$. This is the same principle as maximizing log-wealth.

When the null is simple, the GRO e-variable is given by the likelihood ratio (see testing by betting—simple vs simple) for the proof). For composite nulls $\mathcal{P}$, the GRO e-variable is given by the pseudo likelihood ratio $q/p^{\ast }$ where $p^{\ast }$ is the reverse information projection (RIPr) of $Q$ onto the convex hull of $\mathcal{P}$.