Fusion: Keynes’ Treatise
Ch. XIII The Fundamental theorems of probable inference, with his ‘multiplication theorems’ covering what is now termed data fusion. Keynes develops this in Chs. XV, XVI.
“It seems to have been generally believed by these and other logicians and mathematicians that the probability of two witnesses speaking the truth, who are independent in the sense that there is no collusion between them, is always the product of the probabilities that each of them separately will speak the truth.
The fallacy of such reasoning is easily exposed by a more exact statement of the problem. … “
“It is evident that the cases in which exact numerical measurement is possible are a very limited class, generally dependent on evidence which warrants a judgment of equiprobability by an application of the Principle of Indifference. The fuller the evidence upon which we rely, the less likely is it to be perfectly symmetrical in its bearing on the various alternatives, and the more likely is it to contain some piece of relevant information favouring one of them. In actual reasoning … exact numerical measures will occur comparatively seldom.”
“A good many of the most misleading fallacies in the theory of Probability have been due to a use of the Multiplication Theorem in its simplified form in cases where this is illegitimate.
These fallacies have been partly due to the absence of a clear understanding as to what is meant by Independence. Students of Probability have thought of the independence of events, rather than of the independence of arguments or propositions. …”