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Inductive applications

We provide two simple inductive applications of the main theorem. The predictive probability (15) may be used to determine the probability of a further observation. In order to show the most simple application of the main theorem, we come back to the example of drawing balls from an urn. Using (15) we can determine the probability that the color of the next drawn ball, Xn+1, is j e {1,2,...,d }. In order to do this we must fix the values of pj for all j, that is, the probability of drawing the first ball of color j, and the value of X. Choosing this value amounts to fixing in what measure the drawing of a ball of a color other that j affects the probability of drawing a ball of color j.

Another more realistic example concerns a clinical trial with dichotomous responses. In this case the objective was to evaluate the effectiveness of a drug (6-mercaptopurine) for the treatment of a deadly disease (acute leukemia). Patients were randomized in pairs to receive either the drug or a placebo. For each pair it was recorded which patient stayed longer in remission, that is, without symptoms of the illness. If it was the drug patient, then the treatment was judged a success, S, otherwise a failure, F. There were pairs of patients in the trial, and the results were as follows:

Thus, the drug was better than the placebo with 18 of the 21 pairs.

These are the data. A very natural question the investigators may ask themselves is this: having a pair of patients both suffering from that deadly disease, if the first receives the drug and the second a placebo, what is the probability that the former stays longer in remission? A moment's reflection is sufficient to confirm that in the clinical trial we are considering the hypotheses of the main theorem, that is, regularity, exchangeability and invariance, hold. As a consequence we can apply (15). In the case we are considering the cells are two and the occupation vector is (18,3). With these data (15) becomes

where X denotes a future pair of patients. In order to have a probability value we must choose the values of the parameters. This can be done along the following lines. First, we suppose we have no evidence about the effectiveness of the drug. For this reason we fix pS = pF = 1/2. Second, we do not want our initial probabilities to substantially affect the predictive probability. Hence we chose a small value of X, say 2 or (which is equivalent) we make the relevance quotient at V equal to two thirds. This choice is tantamount to saying that the patient pairs are strongly (positively) correlated: in other words, that the result with one pair affects in a strongly positive way that with a subsequent pair. With these values (17) becomes

It is worth noting that the values of the parameters giving (18) are those that characterize Laplace's celebrated rule of succession. In TP Keynes pays a great deal of attention to this rule, concluding that

it [the rule of succession] is of interest as being one of the most characteristic results of a way of thinking in probability introduced by Laplace, and never thoroughly discarded to this day.

(TP, p. 417)

The relevance quotient is an important tool for solving inductive problems but it is useful in other contexts, too, such as with statistical mechanics, population genetics, and macroeconomic modeling. In these applications negative dependence, as given by negative values of X, may become important. This is actually what happens in statistical mechanics. In the next sections we give some examples of applications in statistical mechanics and macroeconomic modeling: the former because of its importance in the history of science, the latter because we want to show a way of following the route traced by Keynes by using probability in economics.

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