358 Prof. Doukiu on certain Questions relating to 



equal to 1 ; whence the expression given above follows imme- 

 diately. 



Theobem III. — Let H,, Hg, . . . H„ 6e mutually exclusive and 

 exhaustive hypotheses, of which the probabilities relative to a cer- 

 tain state of information are Pj, Pj, . . . P„. If new and indepen- 

 dent information be obtained, in consequence of which their proba- 

 bilities loould have become pi, Pj . . . p„, supposing them to have 

 been all equal at first, then the probability of Hj actually becomes 



2(P,K.)' 



For first suppose the new information to have reference to an 

 independent set of hypotheses K„ Kg, . . . K„ with a piiori equal 

 probabilities. Then the probability of K^ becomes p^, and the 

 probabilities of the combinations (Hj, Kj), (H^, Kg) • • • • become 

 ^\P\y ^'ilh) • • • P Z'- Now the ratios of these probabilities 

 would not be affected by the discovery that (for all values of i) 

 Hj. and K. mutually imply one another, so as to amomit to the 

 same hypothesis; but then, all other combinations being ex- 

 cluded, the sum of the probabilities in question would become 1, 

 Avhence the theorem follows immediately. 



A slight consideration of the form of the expression thus 

 arrived at will suffice to establish a more general theorem, which 

 may be thus stated : — 



Theorem IV. — Let H,, Hg, . . . Hn be mutually exclusive and 

 exhaustive hypotheses, of which the h priori probabilities are equal. 

 If information be obtained fi'oni several independent sources, which, 

 considered separately, ivould give respectively p^, qt, . . . Vj, for the 

 probability o/Hj, then the actual probability o/Hj is 



Mpili-'-^i) 

 9. The theorems just established have their most important 

 applications in the theory of the combination of observations, 

 and generally in the solution of problems respecting the infer- 

 ences drawn from phsenomena. I shall proceed to discuss a few 

 miscellaneous examples, chosen with a view to exhibit the accord- 

 ance between the results of the theory and the conclusions of 



* The reasoning in the text may be comiiarecl with that emj)loye(l by 

 Prof. De Morgan in his pajier " On the Structure of the Syllogism," &c. 

 (Camb. Phil. Trans., vol. viii. part '<i.) See ])articularly pp. 26, 21 . With 

 respect to whatever eoiTcspondence may be considered to e.xist between 

 that paper and the present essay, it will I think be sufficient to say, that 

 the aljove mode of treating the subject was familiar to me before I saw the 

 memoii' in question. 



