Abstract
As we have already mentioned, Bayes’s books and papers were demised — or so one is sometimes given to believe — to the Reverend William Johnson, his successor at the Pantile Shop1 at Little Mount Sion. Timerding [1908] concludes that
nach seinem Ableben betrauten seine Angehörigen Price mit der Durchsicht seiner hinterlassenen Papiere, in denen verschiedene Gegenstände behandelt waren, deren Veröffentlichung ihm aber seine Bescheidenheit verboten hatte [p.44]
but it is difficult to see, on the basis of Bayes’s posthumous publications, why he should have papers on “sundry matters” ascribed to him, and why his not publishing should be attributed (or even attributable) to a modesty2 Miranda might well have envied.
Et his principiis, via ad majora sternitur.
Isaac Newton.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
See Brewer, The Dictionary of Phrase and Fable [1978, p.938].
This pudency (or prudency?) seems first to have been noticed by William Morgan [1815, p.24], who wrote “On the death of his friend Mr. Bayes of Tunbridge Wells in the year 1761 he [i.e. Price] was requested by the relatives of that truly ingenious man, to examine the papers which he had written on different subjects, and which his own modesty would never suffer him to make public.” Hacking [1965, p.201] writes “Cautious Bayes refrained from publishing his paper; his wise executors made it known after his death. It is rather generally believed that he did not publish because he distrusted his postulate, and thought his scholium defective. If so he was correct.” In 1971, however, and writing on this same point, Hacking says of Bayes that “His logic was too impeccable” [p.347]. Stigler [1986a, p.130] suggests that any reluctance Bayes might have felt towards publication could perhaps be attributed to difficulty in the evaluation of the integral of his eighth proposition. Good [1988] mentions three reasons for non-publication, viz. (i) the implicit assumption that a discrete uniform prior for r (the number of successes) implies a continuous uniform prior for p (the physical probability of a success at each trial), (ii) the essential equivalence of the assumptions as to the two priors in (i) when N (the number of trials) is large, and (iii) the first ball (by means of which p is determined) is essentially a red herring.
Canton is described in Pearson [1978] as “the Royal Society Secretary” [p.369]. However his name does not appear in Pearson’s list of secretaries on p.369, nor in Signatures in the First Journal-Book and the Charter-Book of the Royal Society: nor is he listed as holding office in the Royal Society in The Record of the Royal Society of London [1912].
Commenting on this letter, Savage, in an unpublished note of 1960 (printed as the Appendix to the present work), wrote “this is apparently the first notice ever taken of asymptotic series”. On this point see Appendix 1.1 to Chapter 1. Deming (see Molina and Deming [1940, p.xvi]) states that the manuscript was submitted to the Royal Society by Price.
On works attributed to Bayes see Pearson [1978, p.360–361].
For reprints and summaries of the Essay the following should be consulted: Barnard [1958] (reprinted in Pearson and Kendall [1970]: note also comment in Sheynin [1969]), Bru and Clero [1988], Dinges [1983], Edwards [1978], Fisher [1956/1959], Molina [1931], Molina and Deming [1940] (reviewed by Lidstone [1941]), Press [1989] and Timerding [1908] (see Pearson [1978, pp.366, 369]). The 1918 catalogue of the Printed Books in the Edinburgh University Library lists, as number 0*22.14/1, a work entitled “A Method of Calculating the Exact Probability of All Conclusions founded on Induction. By the late Rev. Mr. Thomas Bayes, F.R.S.” I am indebted to Mrs. Jo Currie of the Special Collections section of that library for the information that this work is in fact merely a reprint of the Essay: with it is bound the supplement (listed as O*22.14/2), both being reprinted in this edition in 1764.
Unlike all Gaul.
Price’s nephew, William Morgan, writes: “Among these [i.e. Bayes’s papers] Mr. Price found an imperfect solution of one of the most difficult problems in the doctrine of chances …” [1815, p.24]. Later he speaks of Price as “completing Mr. Bayes’s solution”. It seems clear from Price’s introductory remarks to the essay, however, that the major part of the latter was presented as Bayes had left it, though Price did expand on the Rules given by Bayes.
See de Finetti [1972, p.159].
See Savage [1960]. Hacking [1971] finds in Price’s introduction “perhaps the most powerful statement ever, of the potential relations between probability and induction” [p.347].
Condorcet [1785, p.lxxxiii] traces the idea to Jacques Bernoulli & de Moivre.
Bernoulli’s Law of Large Numbers, in modern terms, runs as follows: let \(\left\{ {{A_i}} \right\}\)be a sequence of independent events with\(\Pr \left[ {{A_i}} \right] = p,\)where i is a natural number. For every ∈ > 0,\(\Pr \left[ {\left| {S{}_n/n - p} \right| \ge \in } \right] \to 0asn \to \infty \) where \( S_n = \sum\nolimits_I^n {I\kappa } \)(Here I k denotes the indicator function of A k , i.e. that function taking on the values 1 on A k and 0 off A k .) Baker [1975, p.162] does not find it surprising that the foundations for the inversion of Bernoulli’s Theorem were laid in England; he traces this to some aspects of Newtonian philosophy. Further he notes (op. cit. p.166) the relationship between Bayes’s passage from a physical model of probability to an epistemological interpretation, and Price’s appendix showing clear evidence of the logic of Hume’s Treatise. For further details on this last point see Gillies [1987].
Price also comments [p.373] on the defects of the asymptotic nature of de Moivre’s results. The respect in which Price held de Moivre is commented on by Morgan [1815, p.39] as follows: “In the first of these papers [the two published in the Philosophical Transactions in 1770] he corrected an error into which M. De Moivre had fallen; … From the high opinion he entertained of the accuracy of De Moivre, he conceived the error to be his own rather than that of so eminent a mathematician, and in consequence puzzled himself so much in the correction of it, that the colour of his hair, which was naturally black, became changed in different parts of his head into spots of perfect white.” See Dale [1988b] for a discussion of the relationship between Bayes’s theorem and the inverse Bernoulli theorem. In the twelfth volume (1763–1769) of the Philosophical Transactions (Abridged) we find the following comment on Bayes’s problem: “In its full extent and perfect mathematical solution, this problem is much too long and intricate, to be at all materially and practically useful, and such as to authorize the reprinting it here; especially as the solution of a kindred problem in Demoivre’s Doctrine of Chances, p.243, and the rules there given, may furnish a shorter way of solving this problem. See also the demonstration of these rules at the end of Mr. Simpson’s treatise on ‘The Nature and Laws of Chance’.” [p.41]. The reference to De Moivre being to his Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem expansi, it appears that there was also some confusion here between Bayes’s result and the inverse Bernoulli theorem.
The existence of this supplement to the Essay is not mentioned by Barnard [1958] (see Sheynin [1969, p.40]), although note is taken of it in the note at the end of the reprint of his article in Pearson & Kendall [1970].
See Sheynin [1969] for further comments and discussion.
Bayes’s formulation of the problem is viewed by de Finetti, a leading subjectivist, as unsatisfactory — see his [1972, p.158].
This note is printed as the Appendix to the present work.
Seven definitions and seven propositions — is their purpose analogous to that of Carroll’s maids and mops?
But see Price’s introduction to the Essay, foot of p.372.
See de Moivre [1756].
Savage [1960] calls it “of course most interesting”. See also Bernoulli’s Ars Congedandi and de Moivre’s The Doctrine of Chances.
See Savage [1960].
Edwards [1974, pp.44–45].
See Fine [1973, pp.60–61], Hacking [1975, pp.152–153] and Shafer [1976].
Perhaps this supports de Finetti’s [1937] view that the idea of “repeated trials” is meaningless for subjective probability — see Kyburg and Smokier [1964, p.102].
See Edwards [1978, p.116] for references.
More correctly, one postulate in two parts.
Some writers, including Pearson [1920a] and Fisher [1959], have referred to Bayes’s table as a billiard table (which of course is not square). One might wonder whether such referral, occurring as it does in connexion with matters of chance, perhaps embodies a pun, as the word “hazard” was formerly used for a pocket of a billiard table.
This specifies a uniform distribution in the plane: the deduction of a uniform distribution over the side of the table is tacit.
Note also Edward’s [1978, p.116] reformulation.
“A deliberately extramathematical argument in defense of Bayes’ postulate”, Savage [1960].
This seems to imply exchangeability: for example, if a coin is tossed three times, the scholium says that Pr [3 heads] = Pr [2 heads] = Pr [1 head] = Pr [0 heads] ( =¼ presumably!), and hence, for example \(\Pr \left[ {HHT} \right] = \Pr \left[ {HTH} \right] = \Pr \left[ {THH} \right] = \frac{1}{{12}}\) See Edwards [1974, p.48] and Zabell [1982].
See Savage [1960].
See Dale [1982] and, for a contrary assertion, Edwards [1978, p.117].
For comment on Bayes’s evaluation of the incomplete beta-integral see Lidstone [1941, pp.178–179], Molina and Deming [1940, pp.xi–xii], Sheynin [1969, p.4], [1971a, p.235], Timerding [1908, pp.50–51] and Wishart [1927]. The last of these authors points out [p.10] an erroneous value given by Bayes and undetected by Timerding. For a detailed study of the incomplete beta-function see Dutka [1981].
See Pearson [1978, p.369].
Notice that this is also framed in terms of ratios of causes — see p.406 of the Essay.
Note the comment by Waring in Todhunter [1865, art.839]. See also Savage [1960], and Pearson [1978, pp.365–366]. It seems that Price, and not Bayes, was perhaps the first to frame a sort of “rule of succession” argument. See Keynes [1921/1973, chap. XXX] for commentary on this rule.
Dinges [1983, p.95] is one of the few authors to acknowledge this problem as being posed by Bayes. The mentioning of events occurring under the same circumstances as they have in the past can perhaps be traced back to G. Cardano (1501–1576), in whose Liber de Ludo Aleae, caput VI, we read “Est autem, omnium in Alea principalissimum, aequalitas, ut pote collusoris, astantium, pecuniarum, loci, fritilli, Aleae ipsius.”
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Dale, A.I. (1991). Bayes’s Essay. In: A History of Inverse Probability. Studies in the History of Mathematics and Physical Sciences, vol 16. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0415-9_2
Download citation
DOI: https://doi.org/10.1007/978-1-4684-0415-9_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-0417-3
Online ISBN: 978-1-4684-0415-9
eBook Packages: Springer Book Archive