The Bayes e-value: Testing by betting given a prior probability of the null hypothesis

The player called ``Skeptic'' bets $$1$ against the simple nullhypothesis that a random sample $X$ will be drawn from a distributionof probability density function $f₀$, knowing that it will otherwisebe drawn from a distribution of probability density function $f₁$.Skeptic also knows the prior probability of the null hypothesis tobe $π₀$, a number in $[0,1]$. In return for the$$1$, Skeptic chooses to receive the payout that is log-optimalaccording to the prior predictive distribution of probability densityfunction f=π₀f₀+(1-π₀)f₁. That payoutis the e-variable E=f(X)/f₀(X). With xas the observed sample, the e-value that realizes E is e=f(x)/f₀(x),where f(x) is known as the marginal likelihood. Consideringe as the degree to which the null hypothesis is disproven resolvescertain pathologies in evidence theory while reflecting the priorprobability of the null hypothesis. To generalize that, let E₍₀₎ denote any e-variablethat tests a simple or composite null hypothesis, and let e₍₀₎be its e-value for $X=x$. The corresponding Bayes e-variable is E_π₀=π₀+(1-π₀)E₍₀₎,and its realization, the Bayes e-value, is e_π₀=π₀+(1-π₀)e₍₀₎. The special case of E₍₀₎=f₁(X)/f₀(X)and the resulting Bayes factor e₍₀₎=f₁(x)/f₀(x)yield E_π₀=E and e_π₀=e, respectively. In another special case, arguably important in many scientific applications,including testing a Bayesian model known to be false, is π₀=0,leading to E_π₀=E₍₀₎ and e_π₀=e₍₀₎.By contrast, the posterior probability of the null hypothesis wouldin that case be 0 regardless of the data, rendering it useless fordata analysis. For π₀>0, as is suitable for many applications in psychology,biomedicine, genetics, and genomics, e_π₀ is regularizedtoward 1 to the extent that the null hypothesis has high prior probability.