AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 395 



(1 + g) (1 + g') (1 -t- u") ... -(!-»)(! - u') (l --^a^')„. 

 (I + a) (1 + a) (1 + O ... + (1 - a) (1 - g) (l - a).,. ' 



If fi =(!-;- (1 - m) &c., we find that the joint relative testimony is the product of the separate 

 relative testimonies ; which is the easiest way of expressing the result. Thus two authorities 

 of 3 and 4 truths to one error, amount to one authority of 12 truths to one error. 



I need hardly say that the preceding conclusions are verified by their giving such results as the 

 following; — that if one of the authorities be absolute, the joint authority is the same; that any 

 number of testimonies, each without authority either way, gives no authority either way ; that 

 inauthoritative testimonies do not afl'ect the authority of the rest ; and so on. 



Problems of the preceding character are usually solved by the inverse method ; or by the 

 determination of the probabilities of precedent states from an observed event. Others have noted, 

 I suppose, what has often struck me, namely, that the arrangement of conditions into an observed 

 event and its precedents, is sometimes made in a very indirect and unnatural manner. There 

 are however two classes of problems which give the .same results : each inverse problem has a direct 

 problem of the other class connected with it. For instance, there are m and m white balls, and 

 n and n black balls, in two urns. A white ball has been drawn ; what is the probability that the 

 first urn was that which held it ? The answer is well known to be 



m{m' + n) divided by iri{ni' + n) + m' (m + n). 



Now take the following problem. The black balls are absolutely fixed in the urns ; and the white 

 balls are so connected that one will come out of neither, except when a white ball is touched in both, 

 which will only set free one, say the one which was touched first. With one hand in each urn, not 

 knowing one from the other, ttie chance of bringing out a white ball from the first urn (if we 

 try until a ball comes from one or the other) is the same as that above, namely, that a ball drawn 

 white was in the first urn. These two problems are really the same; the first says that a white ball 

 has been drawn, the second that a white ball must be drawn. And precisely the same sort and 

 amount of reflexion which must be employed to make this sameness apparent, must also be employed 

 before the problems above alluded to will lose that indirect and unnatural appearance to which 

 I have referred. It should also be noticed, that any problem on an event to come may, by supposino- 

 the event to have happened, not being vet known, be made a problem of inverse probabilities. 



Prob. 2. Supposing the authorities to bias one another, required the method of allowing 

 for the bias. 



When one authority expressly cites and defers to another, he does not thereby diminish his 

 own authority. For what we want to know of him is simply the value of his assent, which, unless 

 we have some specific reason, we have no more right to suppose less than his average when he judges 

 of another, than we have to suppose it greater. And, in fact, there are men who are better authori- 

 ties as to their judgment of others, than as to what tiiey propose themselves. Neither, for a siniilar 

 reason, does it diminish the value of the second authority, that the conclusion asserted never would 

 have been known to him had it not been for the first. What we want to account for here is 

 undue bias, which I define to exist when there is a proportion of the conclusions of the second 

 authority which are no better for his testimony than they would have been if the first alone had 

 asserted them. The case of a number of autliorities would lead to a complicated result. Suppose 

 three, the values of whose testimonies arc /i, /x, /i" \ and let X' and A" be the probabilities that the 

 second and third are unduly biassed by the first. Then the value of the joint testimony is 



A'x'V + (i -x')\"— ^-y, -^'Vrr- ^ + ^'<' '"' '''' 



mi + (I - fi){i - It.) 



+ (I -\')(I -\ )-^„— ^-!^ 



.3E 2 



