Higher Order Probabilities
This addresses a foundational issue in probability theory and belief representation, but it is incremental as it refutes existing claims rather than introducing new methods.
The paper tackles the problem of whether higher order probabilities are necessary for fully specifying belief, showing that they can always be replaced by marginal distributions of joint probability distributions, with no conceptual or computational advantages found.
A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.