The 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 a 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 Bayes e-variable is \(E_{\pi _{0}}=\pi _{0}+\left( 1-\pi _{0}\right) E_{\left( 0\right) }\) , and its realization, the Bayes 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 \(e_{\left( 0\right) }=f_{1}\left( x\right) /f_{0}\left( x\right) \) yield \(E_{\pi _{0}}=E\) and \(e_{\pi _{0}}=e\) , respectively. 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 applications in psychology, biomedicine, genetics, and genomics, \(e_{\pi _{0}}\) is regularized toward 1 to the extent that the null hypothesis has high prior probability.
