Qualitative Robust Bayesianism and the Likelihood Principle

We argue that the likelihood principle (LP) and weak law of likelihood (LL) generalize naturally to settings in which experimenters are justified only in making comparative, non-numerical judgments of the form "$A$ given $B$ is more likely than $C$ given $D$." To do so, we first \emph{formulate} qualitative analogs of those theses. Then, using a framework for qualitative conditional probability, just as the characterizes when all Bayesians (regardless of prior) agree that two pieces of evidence are equivalent, so a qualitative/non-numerical version of LP provides sufficient conditions for agreement among experimenters' whose degrees of belief satisfy only very weak "coherence" constraints. We prove a similar result for LL. We conclude by discussing the relevance of results to stopping rules.