Wes Salmon on the Application of Probability to Single Events
To my knowledge, Wes Salmon discussed this problem most systematically in his 1967 book, The Foundations of Scientific Inference, University of Pittsburgh Press, 1967 (he was a professor at Indiana University, when he wrote this book; and I first read, and learned a lot from, this book as a textbook in a course at the University of Michigan in 1968). Since this problem is quite relevant to Mayo's discussion of Error Statistics, let me review and criticize Wes Salmon's view. But I wish to say, in advance, I have a high opinion of his consistent defence of the frequency theory, although I disagree with him.
[Wes Salmon was killed in an automobile accident April 22, 2001. Our deepest regrets and sorrow. We remember his happy smiles and his warm manner of dealing with our students when he taught here in Kyoto during April-July, 2000. See also Salmon Note.]
In the sections IV and V of this book, Wes states three criteria of adequacy for any interpretations of probability, and examines each interpretation, including the subjective interpretation and the frequency interpretation. Naturally (for Wes), the frequency interpretation is expected to be the only one which survives these criteria. And in the process of this examination, the question of the probability of single events is also discussed.
First, let us see the three criteria.
(a) Admissibility. We say that an interpretation of a formal system is admissible if the meanings assigned to the primitive terms in this interpretation transform the formal axioms, and consequently all the theorems, into true statements. ...
(b) Ascertainability. This criterion requires that there be some method by which, in principle at least, we can ascertain values of probabilities. ...
(c) Applicability. The force of this criterion is best expressed in Bishop Butler's famous aphorism, "Probability is the very guide of life". It is an unescapable fact that we are seeking a concept of probability that will have practical predictive significance. (63-64)
Now, you know that Wes is, like his teacher Hans Reichenbach, a powerful advocate of the frequency interpretation. Although there is some serious difficulty as regards Ascertainability (probability is defined as a limit of relative frequency; so the problem is, how to know the value of this limit, given only finite data?), Wes argues at length that the frequency intepretation can satisfy all the three. Then he comes back to the question of applicability again, this time with respect to applications to single events.
The probability of a given outcome determines what constitutes a reasonable bet. According to the frequency interpretation's official definition, however, the probability concept is meaningful only in relation to infinite sequences of events, not in relation to single events. The frequency interpretation, it is often said, fails on this extremely important aspect of the criterion of applicability.
... According to Reichenbach, the probability concept is extended by giving probability a "fictitious" meaning in reference to single events. We find the probability associated with an infinite sequences and transfer that value to a given single member of it. ... This procedure, which seems natural in the case of the coin toss, does involve basic difficulties. The whole trouble is that a given single event belongs to many sequences, and the probabilities associated with the different sequences may differ considerably. The problem is to decide from which sequence to take the probability that is to be attached "fictitiously" to the single event. (90)
It seems to me that this problem is not the only problem, nor the most serious; rather, the "fictitious" meaning needs explanation and justification, and this looks indeed serious (and as we will see shortly, Wes in fact comes to this question later). But let's follow Wes Salmon's argument. Wes reconstructs the problem as a problem of choosing an appropriate reference class, since probability according to the frequentist may be regarded as a relation between two classes: in the expression "P(B, A)", A is the reference class, and B is the attribute class (notice we have changed his notation---due to Reichenbach---into the standard notation of conditional probability). The attribute class has no problem, since in a single case of betting, for instance, the terms of the bet determines the attribute in question, double-six, heads, or the ace of spades. The question comes down to: the proportion of such an attribute in what class? Thus he says:
The problem of the single case is the problem of selecting the appropriate reference class. (91)
But he adopts a different policy from Reichenbach's on this problem.
Reichenbach said that one should choose the narrowest reference class for which reliable statistics are available. I would say, instead, that the single case should be referred to the broadest homogeneous reference class of which it is a member. ... Statistical relevance is the key concept here. (91)
Roughly, if there is a partition of A into AC and A~C such that the probability of B on AC is different from the probability of B on A, then C is statistically relevant to B; if there are no such C, A is homogenous with respect to B. Wes elaborates this point further, but we will skip it, because the next move is the crucial one for our discussion: the distinction between probability theory itself and the practical rules for its application.
Such a distinction would have been helpful, particularly for the problem of the single case. Reichenbach admits that the meaning of "probability" for the single case is fictitious. He does not offer rules which enable us to assign probability values univocally. ... It would have been better, I think, if he had refused to apply the term "probability" to single events at all, but had instead reserved some other term such as "weight" which he often used for this purpose. We could then say that probability is literally and exclusively a concept that applies to infinite sequences, not to single cases. If we want to find out how to behave regarding a single case, we must use probability knowledge; the problem is one of deciding how to apply such knowledge to single events. Our rules of application would tell us to find an appropriate reference class to which our single case belongs and use the value of the probability in that infinite sequence as the weight attached to the single case. The rule of selecting the broadest homogeneous reference class becomes a rule of application, not a rule for establishing values of probabilities within the theory proper. (93)
The distinction Wes introduces looks reasonable; and his cricicism of Reichenbach looks reasonable too. But I have to say "Wait a minute!" against his transition from "probability" to "weight". You can apply arithmetic to ordinary counting, such as "two apples plus one apple equals three apples", because you know, implicitly at least, that apples do not multiply themselves. But if you put two mice, one male and another female, into the same box and feed them several months, you will not apply the silly formula "one mouse plus one mouse equals two mice"! Arithmetic formulas are here almost irrelevant. Now, to which do you think the probability of single events is closer? The aggregation of apples is expected to have the same structure as what the arithmetic formula says, but a couple of mice, considering their reproductive capacity and the period of feeding, may not be so expected. The probability case looks closer to the mouse case. (See the Figure, summarizing Wes's view and our question.)
According to the frequentist, probability presupposes an infinite sequence, and probability by definition presupposes this mathematical structure. But several bets, taken together, do not share such a structure; and in particular, a single bet is far from sharing such a structure. Then, what is the justification for applying probability theory to a single bet, or a single event, for that matter? Wes may reply "You may repeat similar bets", but I can retort "Even if I bet just on this single case in my life, I can speak of a rational betting ratio; what is this ratio, is it an application of a limiting value of such-and-such sequence? Of course not!" In short, it is quite hard to make sense of even speaking of applications of probability theory to single events, according to the frequency interpretation.
I find the preceding point is fatal to the frequency interpretation as regards its application to single events. Still, Wes has something to say; he may defend the use of "weight" in terms of a long-run success:
If we know the long-run probabilities, a certain type of success is assured by following the methodological rules for handling single events. A given individual deals with a great variety of single events. As an aggregate they may not have much in common. They may be drawn from widely different probability sequences. A man bets on the toss of some dice, he decides to carry his unbrella because it looks a bit like rain, he takes out an insurance policy, he plants a garden, he plays poker, he buys some stock, etc. This series of single cases can, however, be regarded as a new probability sequence made up of the single cases the individual deals with. It is demonstrable that he will be successful "in the long run" if he follows the methodological rules laid down. (96)
Now it seems we have here a sort of circular procedure. The only common property of the events in this "new sequence" seems to be that they come under the same "methodological rules". He distinguished applications from the theory, but throw these applications back into the theory, in order to defend the use of the weight for a single case. Presumably, in order to save the weight for single events (which presupposes probability, you remember), he distinguished application from the theory; but then using a rule of application, he goes back into the theory, and construct a sequence for figuring out a limiting probability of the success (such as winning on a bet) of the use of such weights.
It may be argued that this circularity is innocuous, because Wes is here talking about a success of a rule, and it seems all right to make a rule and take statistics of its applications; it is quite different from assuming A in order to prove A, it may be alleged. But we have to notice that, even if this new sequence has a limiting probability (an asymptotic property), it does not guarantee any "reasonable success" in a finite case (including one's whole life), still less in a single case. Reichenbach's defence (Wes is appealing to this) of inductive rules in terms of "long-run success" does not give any measure of "reliability" (recall that Mayo emphasizes this!) in a short run, as Wes clearly realizes ("no probability is assigned to the statement of the limit of the relative frequency", 95; hence no measure of reliability). Then what is the use of this? What sort of practical guide does it give? This is the same question, all over again, as the original question as regards the use of weights for single cases. Thus, the charge of circularity is not merely apparent, it is indeed genuine.
Therefore, after all, we have to conclude that Wes's maneuvre of introducing the distinction of theory and its application is unhelpful for his defence of the frequency interpretation.
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