The mathematical principles of quantum theory provide a general foundation for assigning probabilities to events. This paper examines the application of these principles to the probabilistic inference problem in which hypotheses are evaluated on the basis of a sequence of evidence (observations). The probabilistic inference problem is usually addressed using Bayesian updating rules. Here we derive a quantum inference rule and compare it to the Bayesian rule. The primary difference between these two inference principles arises when evidence is provided by incompatible measures. Incompatibility refers to the case where one measure interferes or disturbs another measure, and so the order of measurement affects the probability of the observations. It is argued that incompatibility often occurs when evidence is obtained from human judgments.