I recall first that at the outset of all problems of the probability of effects that have hitherto occupied us, we have always had to make a convention, more or less justified. And if in most cases the result was, in a certain measure, independent of this convention, this was only because of certain hypotheses which permitted us to reject _a priori_ discontinuous functions, for example, or certain absurd conventions.
We shall find something analogous when we deal with the probability of causes. An effect may be produced by the cause _A_ or by the cause _B_. The effect has just been observed. We ask the probability that it is due to the cause _A_. This is an _a posteriori_ probability of cause. But I could not calculate it, if a convention more or less justified did not tell me _in advance_ what is the _a priori_ probability for the cause _A_ to come into play; I mean the probability of this event for some one who had not observed the effect.
The better to explain myself I go back to the example of the game of écarté mentioned above. My adversary deals for the first time and he turns up a king. What is the probability that he is a sharper? The formulas ordinarily taught give 8/9, a result evidently rather surprising. If we look at it closer, we see that the calculation is made as if, _before sitting down at the table_, I had considered that there was one chance in two that my adversary was not honest. An absurd hypothesis, because in that case I should have certainly not played with him, and this explains the absurdity of the conclusion.
The convention about the _a priori_ probability was unjustified, and that is why the calculation of the _a posteriori_ probability led me to an inadmissible result. We see the importance of this preliminary convention. I shall even add that if none were made, the problem of the _a posteriori_ probability would have no meaning. It must always be made either explicitly or tacitly.
Pass to an example of a more scientific character. I wish to determine an experimental law. This law, when I know it, can be represented by a curve. I make a certain number of isolated observations; each of these will be represented by a point. When I have obtained these different points, I draw a curve between them, striving to pass as near to them as possible and yet preserve for my curve a regular form, without angular points, or inflections too accentuated, or brusque variation of the radius of curvature. This curve will represent for me the probable law, and I assume not only that it will tell me the values of the function intermediate between those which have been observed, but also that it will give me the observed values themselves more exactly than direct observation. This is why I make it pass near the points, and not through the points themselves.