The marginal e-value: Testing by betting given a prior probability of the null hypothesis
Résumé
The infamous player called ``Skeptic'' bets $\$1$ against the simple null hypothesis that a random sample $X$ will be drawn from a distribution of probability density function $f_{0}$, knowing that it will otherwise be drawn from a distribution of probability density function $f_{1}$. Skeptic also knows the prior probability of the null hypothesis to be $\pi_{0}$, a number in $\left[0,1\right]$. In return for the $\$1$, Skeptic chooses to receive the payout that is log-optimal according to the prior predictive distribution of probability density function $f=\pi_{0}f_{0}+\left(1-\pi_{0}\right)f_{1}$. That payout is the e-variable $E=f\left(X\right)/f_{0}\left(X\right)$. With $x$ as the observed sample, the e-value that realizes $E$ is $e=f\left(x\right)/f_{0}\left(x\right)$, where $f\left(x\right)$ is known as the marginal likelihood. Considering $e$ as the degree to which the null hypothesis is disproven resolves certain pathologies in evidence theory while reflecting the prior probability of the null hypothesis. To generalize that, let $E_{\left(0\right)}$ denote any e-variable that tests a simple or composite null hypothesis, and let $e_{\left(0\right)}$ be its e-value for $X=x$. The corresponding marginal e-variable is $E_{\pi_{0}}=\pi_{0}+\left(1-\pi_{0}\right)E_{\left(0\right)}$, and its realization, the marginal e-value, is $e_{\pi_{0}}=\pi_{0}+\left(1-\pi_{0}\right)e_{\left(0\right)}$. The special case of $E_{\left(0\right)}=f_{1}\left(X\right)/f_{0}\left(X\right)$ and the resulting Bayes factor $e_{\left(0\right)}=f_{1}\left(x\right)/f_{0}\left(x\right)$ degenerates to $E_{\pi_{0}}=E$ and $e_{\pi_{0}}=e$. In another special case, arguably important in many scientific applications, including testing a Bayesian model known to be false, is $\pi_{0}=0$, leading to $E_{\pi_{0}}=E_{\left(0\right)}$ and $e_{\pi_{0}}=e_{\left(0\right)}$. By contrast, the posterior probability of the null hypothesis would in that case be 0 regardless of the data, rendering it useless for data analysis. For $\pi_{0}>0$, as is suitable for many genetics and genomics applications, $e_{\pi_{0}}$ is regularized toward 1 to the extent that the null hypothesis has high prior probability.
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