# Confidence Intervals vs Bayesian Intervals

• E. T. Jaynes
• Oscar Kempthorne
Chapter
Part of the The University of Western Ontario Series in Philosophy of Science book series (WONS, volume 6b)

## Abstract

For many years, statistics textbooks have followed this ‘canonical’ procedure: (1) the reader is warned not to use the discredited methods of Bayes and Laplace, (2) an orthodox method is extolled as superior and applied to a few simple problems, (3) the corresponding Bayesian solutions are not worked out or described in any way. The net result is that no evidence whatsoever is offered to substantiate the claim of superiority of the orthodox method.

To correct this situation we exhibit the Bayesian and orthodox solutions to six common statistical problems involving confidence intervals (including significance tests based on the same reasoning). In every case, we find that the situation is exactly the opposite; i.e., the Bayesian method is easier to apply and yields the same or better results. Indeed, the orthodox results are satisfactory only when they agree closely (or exactly) with the Bayesian results. No contrary example has yet been produced.

By a refinement of the orthodox statistician’s own criterion of performance, the best confidence interval for any location or scale parameter is proved to be the Bayesian posterior probability interval. In the cases of point estimation and hypothesis testing, similar proofs have long been known. We conclude that orthodox claims of superiority are totally unjustified; today, the original statistical methods of Bayes and Laplace stand in a position of proven superiority in actual performance, that places them beyond the reach of mere ideological or philosophical attacks. It is the continued teaching and use of orthodox methods that is in need of justification and defense.

## Keywords

Common Sense Bayesian Method Bayesian Solution Bayesian Test Ancillary Statistic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© D. Reidel Publishing Company, Dordrecht-Holland 1976

## Authors and Affiliations

• E. T. Jaynes
• 1
• Oscar Kempthorne
• 2
1. 1.Dept. of PhysicsWashington UniversitySt. LouisUSA
2. 2.Statistical LaboratoryIowa State UniversityUSA