Jeffrey’s rule of conditioning with various forms for uncertainty

•We extend Jeffrey’s rule of conditioning by allowing alternative formulations for the uncertainty about conditioning variable.•We consider the case where our uncertainty is expressed in terms of a measure.•We look at case where our uncertainty about conditioning variable is expressed in terms of Dempster–Shafer belief structure.

We first introduce Jeffrey’s rule of conditioning and explain how it allows us to determine the probability of an event related to one variable from information about a collection of conditional probabilities of that event conditioned on the state another variable. We note that in the original Jeffrey paradigm we have the uncertainty about the state of the conditioning variable expressed as a probability distribution. Here we extend this by allowing alternative formulations for the uncertainty about the conditioning variable. We first consider the case where our uncertainty is expressed in terms of a measure. This allows us to consider the case where our uncertainty is a possibility distribution. We next consider the case where our uncertainty about the conditioning variable is expressed in terms of a Dempster–Shafer belief structure. Finally we consider the case where we are ignorant about the underlying distribution and must use the decision maker’s subjective attitude about the nature of uncertainty to provide the necessary information to use in the Jeffrey rule.