Some Properties of Bayesian Orderings of Experiments

Abstract

Let $\Omega$ be a finite set of states and $\Xi$ the class of prior distribution on $\Omega$. A nonnegative, continuous, concave function on $\Xi$ is called an uncertainty function and if $X = (\mathscr{X}, \mathscr{A}; \mathbf{P}_\theta, \theta \in \Omega)$ and $Y = (\mathscr{y}, \mathscr{B}; \mathbf{Q}_\theta, \theta \in \Omega)$ are two experiments $X$ is called at least as informative as $Y$ with respect to $U$ if $U(\xi |X) \leqq U(\xi | Y)\quad \text{for all} \xi \in \Xi$ where $U(\xi \mid X)$ is the expected posterior uncertainty for an observation on $X$ when the prior is $\xi \in \Xi$. Any such $U$ induces a partial ordering on the class of all experiments. The paper characterizes (i) the class of functions $U$ which lead to a total ordering of the class of experiments and (ii) the class of transformations of a function $U$ which preserve its induced ordering.