1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

VIOscar SheyninOn the Bernoulli Law of Large NumbersBernoulli considered (independent) trials with a constant probabilityof success, and rigorously proved that the frequency of success tendsto that probability. Mises, however, treated collectives, totalities ofphenomena or events differing from each other in some indication, andcharacterized by the existence of the limiting frequency of success andby irregularity. The latter property meant that for any part of thecollective that limiting frequency was the same.Alimov noted that artificially constructed collectives proved that theempirical frequency of success can become more stable as the numberof trials increased, but have no limit. Therefore, the existence of thatlimit is an experimental fact. I have described his viewpoint in somedetail in an Introduction to [v]. Tutubalin largely sided with Alimov.In the same Ars Conjectandi, previous to proving the LLN,Bernoulli stated that his law was also valid in its inverse sense (and DeMoivre independently stated the same with respect to the first versionof the CLT proved by him in 1733). In other words, an unknown andeven a non-existing probability (one of Bernoulli’s examples) could beestimated by the limiting frequency.In a little known companion paper (1765) to his main memoir(1764), Bayes all but proved his own limit theorem explicating thatinverse LLN. He did not make the final step from the case of a largefinite number of trials because he opposed the application of divergentseries which was usual in those times. That was done in 1908 byTimerding, the Editor of the German translation of Bayes, certainlywithout using divergent series.Bayes – Timerding examined the behaviour of the centred andnormed random variable η, the unknown probability, (η − Eη)/var ηwhereas the direct LLN dealt with the frequency ξ, (ξ − Eξ)/varξ. Hismain memoir became widely known and for a long time the Bayesapproach had been fiercely opposed, partly because an unknownconstant was treated as a random variable (with a uniformdistribution). Note that varη > varξ which is quite natural sinceprobability is only unknown in the inverse case. For attaining the sameprecision the inverse case therefore demands more trials than the directlaw. Mises could have called Bayes his main predecessor; actually,however, he only described the work of the English mathematician,and inadequately at that. Bayes completed the first stage of thedevelopment of probability theory.Alimov’s viewpoint was largely correct since he considered anincomparably more general pattern than Bernoulli and thought aboutthe necessary checks, but he [iv] was too radical in denying importantparts of mathematical statistics as also too brave in altering the Misesapproach. To borrow an expression from Tutubalin [end of ii], heintroduced the Mises approach of a light-weighted type.139