Professor Broad on the Limit Theorems of Probability Max Black Mind, New Series, Vol. 56, No. 222. (Apr., 1947), pp. 148-150. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28194704%292%3A56%3A222%3C148%3APBOTLT%3E2.0.CO%3B2-T Mind is currently published by Oxford University Press.
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http://www.jstor.org Sun Jun 17 04:54:40 2007
PROFESSOR BROAD
ON THE LIMIT THEOREMS OF PROBABILITY.
1. Professor C. D. Broad's derivations of the direct and inverse Bernoulli theorems (and certain corollaries ) present some difficulties by which other readers than myself may have been troubled. Since the interpretation of such " limit theorems " is notoriously difficult, and has been responsible for much misunderstanding in the past, it has seemed worth while to add the following marginal comments to Broad's exposition. 2. The point of deriving the limit theorems from a set of six postulates (given by von Wright, whose work Broad is here reporting and improving) is to provide a " formal analysis " of the theorems, i.e., to show exactly what assumptions are needed to ensure their truth. (When this is done, it will be correspondingly easier to see how far the principles of induction can be justified a posteriori.) The sole test of the merit of the performance must accordingly be the rigour of the demonstration ; and Broad takes pains to fill in the details of von Wright's outline proofs. This procedure has brought to light one gap in the chain of argument (see paragraph 7 below ) ; but I think there are also other respects in which the arguments, even as emended by Broad, stand in need of further improvement. 3. Broad's proof of the " Inverse Principle of Great Numbers " (pp. 109-112) will sufficiently illustrate my misgivings, though similar points could be made about other proofs given by him. The principle in question concerns the deductions concerning the a priori probability of a single success, which may be drawn from knowledge of a constant ratio of successes in an indefinitely extended series of repeated trials. Broad's preliminary and informal explanation of the principle (p. 109) misleadingly suggests that only such a priori probabilities as are measured by rational numbers need to be considered. I say this is misleading because von Wright's axioms in no way restrict probabilities to be rational numbem2 But if the a priori probabilities may form a continuous distribution, rule k at the foot of page 109 is wrong ; for all we need to assume is that the probability density shall not become zero a t the value p. In general, we shall neither know nor need to know that the probability of the event's having an a priori probability p is not zero. 4. This criticism is perhaps not serious ; but I think the next one is. Broad's proof amounts to evaluating a mathematical limit from This JOURNAL, vol. LIII, pages 98-119. See, for instance, axiom (n) on page 98, where real numbers are explicitly admitted as values of probabilities.
first principles ; but his ingenious improvisation implicitly assumes that the quantity 6 is rationa1.l Since his argument ought to hold for " any pre-assigned " quantity 6,2 the proof is defective.3 5. Finally, the appeal to the " Direct Principle of Great Numbers " (middle of p. 111) seems to be mistaken. For that principle supplies a value not for the probability of the proposition f N ( R ; Q) = p as Broad needs ; but rather for the probability of the different proposition f N ! R ; &) & p & 6 (see the symbolic expression for the " Direct Principle " a t the foot of p. 103). 6. If any of these criticisms should prove t o be mistaken, I hope they may a t least illustrate the difficulty, for even the most careful reader, of verifying that Broad has given rigorous proofs of the theorems in which he is interested. ,4nd I think these difficulties arise because he is trying to prove what are essentially theorems of pure mathematics (more specifically those arising in the evaluation of limiting or asymptotic values of given functions) without recourse to the standard mathematical symbolism of limits and integrals. (pp. 114-119) uses a lemma, which Broad finds plausible, but is 7. The proof of the " Statistical Principle of Great Numbers " unable to prove (foot of p. 115). The lemma, which I shall cite as L, asserts the logical equivalence of two propositions now to be described. Suppose some event has an initial and constant (or a priori) probability of occurrence p. Then, on the assumption of the relative independence of successive trials, the following propositioil is true : for given 6, the probability that the relative frepuency of occurrence of the event i n N successive trials shall l;e between p - 6 and p 6 converges to 1 as N increases. This proposition, (y) say, is what Broad calls the " Direct Principle of Great Numbers ". Now let us follow Broad in calling (a) the proposition : the probability that the relative frepuency of occurrence of the event in a n in$nite series of successice trials shall be p i s 1. Now the lemma needed (L) is that (y) is " logically equivalent " to (u) (top of p. 117). 8. If Broad intends that (u) can be deduced from (y) without appeal to other propositions, we should not need to take account of the specific formula (" Newton's formula ") by .which the relative frequency of success in N independent trials is calculated from the initial probability p. Any function exhibiting the kind of " average convergence " over an interval, asserted in (y), would also converge " a t p ", as asserted in (a).
+
See the third line of paragraph 3 on page 110. Since 2 ~ is6 an integer, and p is an integer, 6 must be rational. a Page 110, line 9. 1 suppose one could approximate to irrational values of 6 by means of rational numbers and so, with more or less ingenuity, patch up the argument. But it hardly seems worth the trouble. Since we are to calculate a limiting value, why not use mathematical symbolism as an insurance against mistakes ?
The question to be decided could then be expressed in some such abstract form as the following : Suppose we have some variable function v(x, n), where n is an integer, and suppose we know
1
03
(i) that
v(x, n)dx = 1, for each n ;
-a)
J n-+m
p+d
(ii) that lim
v(z, n)dx = 1for some fixed p and every S ;
P-8
does i t then follow (iii) that
jPt8 lim v(z, n)dz
= 1for
all S ?
'
[-03
P-8
9. It is plausible that (iii) is a stronger condition than (ii) and i t
can in fact be shown that this is the case.2 The situation, then, is that on this interpretation of what Broad means by the " logical equivalence " of (y) and ( a ) ,his conjecture is false. (y), or rather its abstract amalogue, can be deduced from the abstract analogue of (u), but not vice vcrsa. 10. Nevertheless, (u) can be derived from the usual axioms of probability thcory, though not in a trivial or obvious way.3 So the " Statistical Principle of Great Numbers " is a correct assertion. 11. The moral would seem to be that in providing a "formal analysis " of theorems in probability i t would be advantageous to provide a system of axioms which would allow us as quickly as possible to use the full resources of the infinitesimal calculus. It might even then appear that the mathematicians had already succeeded in providing the relevant calc~lations.~ MAX BLACK. Here (i) ensures that v(x, n) shall be a probability distribution, (ii) is the abstract formulation of the direct law of great numbers and (iii) is the analogue of (cc). I refrain from giving a formal proof of this. The reader who is interested in the mathematics of the question may consult H. Cram&, Random Variables (1937) on "convergence in probability" (p. 38 and p. 43) or Kaczmarz and Steinhaus, TAeorie der Orthogonalreihen (1935), page 129. It is a direct consequence of the so-called Cantelli's Theorem (a "stronger" theorem than Bernoulli's) for whose proof see, for instance, Uspensky, Introduction to Bathematical Probability. All of the "limit theorems " are proved by rigorous mathematical methods in R. von Rfises' Wahrscheinlichkeitmechnung. The fact that von Mises subscribes to a frequency interpretation of probability does not affect the correctness of his proofs since he very soon derives the customary axioms of elementary probability theory from his own philosophical principles, and makes no further use of the latter.
