The a Posteriori Calculation of Probability

THE A POSTERIORI CALCULATION OF PROBABILITY In many and most important cases, it is impossible to calculate the probability a priori, because the alternative possi bilities cannot be regarded as equally likely. Take, for instance, the case of a sick man. What is the probability of his recovery from his illness? This cannot be calculated a priori. It is true, of course, that there are only two alternatives—either he will recover or he will not. But these two alternatives cannot be regarded as equally likely, without special evidence. If they could be so regarded, why then the chance of recovery from cerebral meningitis would be the same as that of recovering from a common cold—which is beli0 by actual results. And if the alternative possibilities are not equally likely, as indeed they are not, then the probability cannot be estimated a priori. How, then, shall it be calculated? Well, it can still be calculated a posteriori if the results of a sufficiently large number of cases have been observed, that is to say, if we know the frequency of the type of case under consideration. For then, just as it is possible, as we have seen it is possible, to derive the frequency of an event from its probability, where the probability can be calculated a priori, so it is possible by a reverse process to derive the probability of an event from its observed frequency, even in cases in which the probability cannot be obtained a priori at all. Suppose, for instance, that, in the above-mentioned case of the die, face six did not in the long run appear in approximately one out of six throws, but, say in one out of nine throws. Then by treating this frequency as an index to its probability we should say that the probability of face six turning up is //p, which means that it behaves as if it had an a priori probability of 1/9, or as if it had been one of nine equally likely possibilities—which, of course, it is not. In this way, provided we have sufficient data to determine their frequency, the calculus of probability can be applied to all sorts of otherwise incalculable events, such as birth, marriage, death, and the thousand and one ills that flesh is heir to.

Writers on probability are frequently inclined to regard the a posteriori method of calculating probability as the fundamental method. They would either like to banish a priori calculations altogether or only tolerate them as more or less intelligent antici pations, or frequencies. But this theory (known as the frequency theory) of probability is hardly tenable. Frequencies show con siderable variations with the number of cases observed, e.g., when tossing a coin the proportion of heads to tails varies remarkably, accordingly as one stops at the tooth, i,000th, io,000th or 2o,000th toss. One might obtain almost any proportion by stop ping at the right moment. Hence the need of the saving clause "in the long run"—in the long run a die will throw six in one of the six throws, and so on. Even so the element of arbitrariness is not entirely disposed of. As here conceived, the form of the calculus of probability is the a priori form, of which, as already explained, the a posteriori method is simply an inverse process, which treats frequency as a measure of probability, although the two are really different things. There is nothing

unusual in the use of the inverse process, or the resort to "as if" fictions. Our view of the calculable cases of probability makes it possible to keep together all types of probability without any artificiality or straining. In all cases alike the uncertainty arises from the presence of other possibilities than those contemplated.

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some cases these other possibilities can be allowed for by some direct or indirect calculation; in other cases they cannot be esti mated at all, except in a very rough and unpractical manner. On the other hand, the frequency theory of probability does not really apply to non-measurable cases. Even in its modified form, in which frequency is taken to mean the truth-frequency of cer tain classes of propositions, it seems unsatisfactory. One is asked to determine the probability of a judgment by ascertaining the probability of the class of propositions to which it belongs. But suppose the proposition cannot be assigned to its proper class unless its probability is known? The Use of the Calculus of Probabilities.—One may ask, in conclusion, what is the practical use of calculations of proba bility? Some people have exaggerated ideas on the subject. The exaggeration is due in great part to the common confusion of probability with frequency. Frequencies, when treated with the necessary precautions, may be of great practical value when we are dealing with large numbers of facts of the same kind. This is evident from their use in connection with all varieties of insurance schemes, etc., in which the certainty of large numbers can be relied upon to atone, in some ways, for the uncertainty in the lot of the individual. But whereas frequencies are always concerned with large groups, or with long series of events, or with what happens "in the long run," probability is also concerned with individual cases, or small groups of events. This makes all the difference, as may be seen, say, in roulette. The bank, doing business with a great many players, can rely on frequencies. The individual player, limited to a comparatively small number of hazards, relies on the ambiguous calculus of probability (when he is not guided by sheer superstition). The calculus is always right, even when the player loses. For the actual events are matters of frequency, not of mere probability. 'There are ingenious gambling systems based on frequencies ; but even these systems have their day and cease to be. The best of them depend on "the long run," which easily outruns the resources of the average individual. For similar reasons, even in legitimate insurance business, the company has a great advantage over the individual client. But the practical exigencies of life induce responsible individuals to prefer high risks for comparatively small amounts rather than small risks for large amounts. No mere calculation can eliminate the uncertainty of the probable when individual cases are involved. Here, in the last resort, the only safe, or least unsafe, method of ascertaining its probability consists in a close examination of the actual conditions by a suitable expert. Even life insurance companies probably put more faith in the medical report on each case than in their statistical life-tables.