Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Uncertainty, Information, and Sequential Experiments

M. H. DeGroot
The Annals of Mathematical Statistics
Vol. 33, No. 2 (Jun., 1962), pp. 404-419
Stable URL: http://www.jstor.org/stable/2237520
Page Count: 16
  • Read Online (Free)
  • Download ($19.00)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Uncertainty, Information, and Sequential Experiments
Preview not available

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 Ω denote the set of all possible values of the parameter θ, and suppose that the experimenter's knowledge about the true value of θ can be expressed, at each stage of experimentation, in terms of a probability distribution ξ over Ω. Each distribution ξ indicates a certain amount of uncertainty on the part of the experimenter about the true value of θ, and it is assumed that for each ξ 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 Ω 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 θ 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.

Page Thumbnails

  • Thumbnail: Page 
404
    404
  • Thumbnail: Page 
405
    405
  • Thumbnail: Page 
406
    406
  • Thumbnail: Page 
407
    407
  • Thumbnail: Page 
408
    408
  • Thumbnail: Page 
409
    409
  • Thumbnail: Page 
410
    410
  • Thumbnail: Page 
411
    411
  • Thumbnail: Page 
412
    412
  • Thumbnail: Page 
413
    413
  • Thumbnail: Page 
414
    414
  • Thumbnail: Page 
415
    415
  • Thumbnail: Page 
416
    416
  • Thumbnail: Page 
417
    417
  • Thumbnail: Page 
418
    418
  • Thumbnail: Page 
419
    419