Fact-finding by betting: Resolution of the Blue Bus, Gatecrasher, and Prison Riot paradoxes of proof without neglecting base rates

The degree of proof needed to meet an evidentiary standard is most commonly quantified as the posterior odds or, equivalently, the posterior probability, of the fact in dispute. For example (Pardo, 2023), the preponderance of evidence may require odds above 1:1 (probability above 50%), clear and convincing evidence may require odds above 3:1 (probability above 75%), and proof beyond a reasonable doubt may require odds above 9:1 (probability above 90%). However, that way to quantify the degree of proof generates paradoxes when the evidence is essentially limited to the “naked statistical evidence” of base rates, taken as prior probabilities, as classically illustrated by the Blue Bus case, the Gatecrasher case, and the Prison Riot case. A simpler measure of the degree of proof that is not subject to those paradoxes is the likelihood ratio. Unfortunately, it only bypasses those paradoxes because it is almost completely blind to base rates, making it particularly unreliable in res ipsa loquitur cases and in cases involving DNA matching evidence. To overcome both problems, this paper, drawing on an increasingly adopted approach to statistical hypothesis testing, proposes quantifying the degree of proof by the payout the fact-finder would receive for placing a hypothetical bet against the defendant's hypothesis that the disputed fact is false.