numeraire e-variable

A fundamental result when testing composite nulls in the setting of game-theoretic hypothesis testing.

Given any composite null $\mathcal{P}$ against any point null $Q$, there exists a unique e-value (i.e., e-variable) $X^{\ast }$ such that

$${\mathbb{E}}_Q[X/X^{\ast }]\le 1,\text{for all e-variables }X\text{ for }\mathcal{P}.$$

This always exists. Note that $X^{\ast }$ is an e-variable under $\mathcal{P}$, not under $Q$. Some consequences of this definition:

Log-optimality

The numeraire is the log-optimal e-value under $Q$: ${\mathbb{E}}_Q[\log (X/X^{\ast })]\le 0$. This means it is in some sense the best e-value when testing against the alternative $Q$, as its log-wealth grows the fastest (see maximizing log-wealth and growth rate conditions in sequential testing).

Reverse Information Project (RIPr)

Define the (sub)-measure $P^{\ast }$ by $d{P}^{\ast }/dQ=1/X^{\ast }$ for numeraire $X^{\ast }$. $P^{\ast }$ satisfies

$${\mathbb{E}}_Q[\log X^{\ast }]=\underset{X\in \mathcal{E}}{\sup }{\mathbb{E}}_Q[\log X]=\underset{P\in {\mathcal{P}}^{\circ \circ }}{\inf }H(Q\Vert P)=H(Q\Vert P^{\ast }),$$

i.e., it is the reverse information projection (RIPr). Here $\mathcal{E}$ is the set of e-variables for $\mathcal{P}$. So the first equality says that again that $X^{\ast }$ is log-optimal among all e-variables. ${\mathcal{P}}^{\circ \circ }$ is the bipolar of $\mathcal{P}$, which is the set of measures $R$ such that ${\mathbb{E}}_R[X]\le 1$ for all $X\in \mathcal{E}$. ${\mathcal{P}}^{\circ \circ }$ is the effective null hypothesis, i.e., all distributions where the e-values for $\mathcal{P}$ will also be e-values, so in some sense we won’t be able to tell $\mathcal{P}$ and ${\mathcal{P}}^{\circ \circ }$ apart. The final inequality says that $P^{\ast }$ achieves the minimum relative entropy wrt $Q$, which is also the optimal growth rate.