Posterior previsions for the parameter of a binomial model via natural extension of a finite number of judgments

In this paper, we investigate the use of assessments of conditional previsions for modeling prior information on the parameter of a binomial model as a way of obtaining non-vacuous posterior previsions via natural extension. More specifically, we argue that a useful method for obtaining an imprecise prevision for the parameter θ of a binomial model, given a sample of size n showing r successes, is to assess imprecise previsions for θ which are conditional on samples having sizes larger than n. Inferences obtained using this approach are compared to Walley's proposal for learning from a bag of marbles.