Higher Order Probabilities

TL;DR

The paper demonstrates that higher order probabilities can be replaced by joint probability distributions, showing they offer no conceptual or computational advantages over standard probabilities.

Contribution

It proves that higher order probabilities are unnecessary by replacing them with joint distributions, simplifying the conceptual framework of belief representation.

Findings

Higher order probabilities can be replaced by joint distributions.

No advantages of higher order probabilities in concept or computation.

Simplifies the formal understanding of belief systems.

Abstract

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.