Uncertainty, Information, and Sequential Experiments

Abstract

Consider a situation in which it is desired to gain knowledge about the true value of some parameter (or about the true state of the world) by means of experimentation. Let $\Omega$ denote the set of all possible values of the parameter $\theta$, and suppose that the experimenter's knowledge about the true value of $\theta$ can be expressed, at each stage of experimentation, in terms of a probability distribution $\xi$ over $\Omega$. Each distribution $\xi$ indicates a certain amount of uncertainty on the part of the experimenter about the true value of $\theta$, and it is assumed that for each $\xi$ this uncertainty can be characterized by a non-negative number. The information in an experiment is then defined as the expected difference between the uncertainty of the prior distribution over $\Omega$ and the uncertainty of the posterior distribution. In any particular situation, the selection of an appropriate uncertainty function would typically be based on the use to which the experimenter's knowledge about $\theta$ is to be put. If, for example, the actions available to the experimenter and the losses associated with these actions can be specified as in a statistical decision problem, then presumably the uncertainty function would be determined from the loss function. In Section 2 some properties of uncertainty and information functions, and their relation to statistical decision problems and loss functions, are considered. In Section 3 the sequential sampling rule whereby experiments are performed until the uncertainty is reduced to a preassigned level is studied for various uncertainty functions and experiments. This rule has been previously studied by Lindley, [8], [9], in special cases where the uncertainty function is the Shannon entropy function. In Sections 4 and 5 the problem of optimally choosing the experiments to be performed sequentially from a class of available experiments is considered when the goal is either to minimize the expected uncertainty after a fixed number of experiments or to minimize the expected number of experiments needed to reduce the uncertainty to a fixed level. Particular problems of this nature have been treated by Bradt and Karlin [6]. The recent work of Chernoff [7] and Albert [1] on the sequential design of experiments is also of interest in relation to these problems.