By Roy Harrod (auth.)
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Additional info for Foundations of Inductive Logic
It may be useful to give a very simple example of an application of the Bayes theorem. Let there be a manufacturer of dice - it is painful to mention thus early those little pieces which have done so much to bedevil logic - who has done honest trade but also entered into contracts with fraudulent customers for biased dice. Let it be known that ordinary dice, defined as true dice, show a six on average once in every six throws. Let the biased dice be cunningly devised so that on a run of throws the six will show on average once in every two throws.
But the probabilities pertaining to the hypotheses themselves require some prior knowledge, and often, as in our example, a great deal. If the preliminaries of our illustration seemed a little far-fetched, it may be that one has to go a long way to find a suitable instance. Let us suppose that all the information about the makers and the markings did not exist, and that there was no relevant information about the prior probability of bias. What would then be the significance of a run of sixes ?
Since probability is rightly defined as a relation between evidence and a conclusion, and since the m value belongs to statements taken in isolation, the m value is not called a probability. But in its logical aspect, that is, in terms of thought and not of symbols, this m• function is an initial prior probability. It would be premature and unfair to raise the doubt at this stage whether the postulate of equal initial prior probabilities for every structural description would enable the inverse theorem to yield factual probabilities of interesting value ; there would be terrific factorial numbers in the denominator of the initial prior probability.