848 PROCEEDINGS OP THE AMERICAN ACADEMY 



XXIII. 



ON PEIRCE'S CRITERION. 

 By Benjamin Peirce. 



My DEAR Sir, — I perceive that the theory of my criterion has 

 been frequently misunderstood. I presume this to be due in a great 

 degree to the conciseness of the argument with which it was published ; 

 and I propose to remedy this defect. 



The problem which I undertook to solve was the following. There 

 being given certain observations, of which the greater portion is to 

 be regarded as normal and subject to the ordinary law of error adopted 

 in the method of least squares, while a smaller unknown portion is 

 abnormal and subject to some obscure source of error, to ascertain the 

 most probable hypothesis as to the partition of the observations into 

 normal and abnormal. The princii>le adopted in my solution of the 

 problem is the universally recognized doctrine that the measure of the 

 probability of an hypothesis compared with other hypotheses equally 

 probable in other respects is the probability that the event will occur 

 under the hypothesis, and that the most probable hypothesis is that 

 under which the event is most probable. This is the literal expres- 

 sion of the mathematical analysis published in Gould's Astronomical 

 Journal for 1852. The Criterion lias been used otherwise than in the 

 Coast Survey, and especially by my friend Dr. Gould himself. Dr. 

 Gould's tables have greatly facilitated its use, and his sound judgment 

 given in favor of its validity is at least as valuable as that of any living 

 geometer. It has also been much used by that excellent authority Mr. 

 Schott, as in a letter hereto appended. 



The evidence, by which certain observations are placed in the doubt- 

 ful list and subjected to scrutiny, whether they should be rejected, 

 must be exclusively the magnitude of the errors which they involve, 

 when these errors are computed as if they were normal observations. 

 This would not seem to be obnoxious to the charge of inconsistency, 

 any more than is the ordinary Rtdnctio ad Absurdum, in using a 

 method as correct in an observation where it was finally rejected. An 



