This chapter introduces Bayes’ theorem and uses it to ask critical questions about the diagnosticity of evidence – its ability to distinguish between competing hypotheses. It applies this framework to two controversies: whether vaccines cause autism and whether Biden legitimately won the 2020 US election. In both cases, one side relied exclusively on nondiagnostic evidence, facts that have alternative explanations. A key focus is the likelihood ratio, a term in Bayes’ theorem that can represent how much trust we should place in various sources of evidence (e.g., scientific experts, election officials) and in processes like peer or judicial review. The chapter critically examines the peer review system through the case of Andrew Wakefield’s retracted study linking vaccines to autism. Peer review failed to detect misreporting of data and a conflict of interests – highlighting the need to supplement peer review with scientific replication. Together, these examples underscore the importance of public understanding of epistemic institutions and the use of deliberative argumentation to explore alternative hypotheses.