Imprecise Bernoulli Processes

In classical Bernoulli processes, it is assumed that a single Bernoulli experiment can be described by a precise and precisely known probability distribution. However, both of these assumptions can be relaxed. A first approach, often used in sensitivity analysis, is to drop only the second assumption: one assumes the existence of a precise distribution, but has insufficient resources to determine it precisely. The resulting imprecise Bernoulli process is the lower envelope of a set of precise Bernoulli processes. An alternative approach is to drop both assumptions, meaning that we don’t assume the existence of a precise probability distribution and regard the experiment as inherently imprecise. In that case, a single imprecise Bernoulli experiment can be described by a set of desirable gambles. We show how this set can be extended to describe an imprecise Bernoulli process, by imposing the behavioral assessments of epistemic independence and exchangeability. The resulting analysis leads to surprisingly simple mathematical expressions characterizing this process, which turn out to be the same as the ones obtained through the straightforward sensitivity analysis approach.