The Annals of Mathematical Statistics

Bayes, Likelihood, or Structural

D. A. S. Fraser

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Abstract

The traditional model of statistics is examined in Section 1. The model, as such, distinguishes response values only by their likelihood functions. Recognition of this restricted identification effectively precludes the need for any theory of sufficient statistics. A probability space that has an attached observer-processor-reporter (OPR) mechanism is examined in Section 2 as a means of assessing the nature of reported information; such information may or may not be observational in character depending on properties of the OPR mechanism. A variation-response model is a probability space and a class of random variables: the probability space describes the sources of variation in the system under investigation and the class of random variables describes the possible presentations of this variation in the response of the system. Section 3 examines how realized values on the probability space are distinguished or identified by the model; Section 4 considers how distributions on the probability space are identified by response variable data. In Sections 5, 6, 7 the essentials of three contemporary approaches to inference are presented and each is accompanied by criticisms that proponents of the other methods might make. The key to these criticisms lies primarily in whether hypothetical information is added to or substantiated information is omitted from the assembled information concerning the system under investigation. In certain contexts the bayesian approach makes an arbitrary but typically consistent choice of input to its analyses; it is not the input suggested by standard invariance analysis. In those cases where a variation-response model is appropriate, the use of this more embracing model presents theoretical support for the bayesian choice (Section 8); but of course with the more embracing model the bayesian premises are not needed to obtain the usual bayesian result. An example in Section 9 illustrates the theoretical simplicity of classical probabilities for certain unknowns other than realized values on a probability space.

Article information

Source
Ann. Math. Statist., Volume 43, Number 3 (1972), 777-790.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692544

Digital Object Identifier
doi:10.1214/aoms/1177692544

Mathematical Reviews number (MathSciNet)
MR307394

Zentralblatt MATH identifier
0281.62005

JSTOR
links.jstor.org

Citation

Fraser, D. A. S. Bayes, Likelihood, or Structural. Ann. Math. Statist. 43 (1972), no. 3, 777--790. doi:10.1214/aoms/1177692544. https://projecteuclid.org/euclid.aoms/1177692544


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