# Bayes, Thomas

# Bayes, Thomas

Thomas Bayes (1702–1761) was the eldest son of the Reverend Joshua Bayes, one of the first nonconformist ministers to be publicly ordained in England. The younger Bayes spent the last thirty years of his comfortable, celibate life as Presbyterian minister of the meeting house, Mount Sion, in the fashionable town of Tunbridge Wells, Kent. Little is known about his personal history, and there is no record that he communicated with the well-known scientists of his day. Circumstantial evidence suggests that he was educated in literature, languages, and science at Coward’s dissenting academy in London (Holland 1962). He was elected a fellow of the Royal Society in 1742, presumably on the basis of two metaphysical tracts he published (one of them anonymously) in 1731 and 1736 (Barnard 1958). The only mathematical work from his pen consists of two articles published posthumously in 1764 by his friend Richard Price, one of the pioneers of social security (Ogborn 1962). The first is a short note, written in the form of an undated letter, on the divergence of the Stirling (de Moivre) series ln(z!). It has been suggested that Bayes’ remark that the use of “a proper number of the first terms of the … series” will produce an accurate result constitutes the first recognition of the asymptotic behavior of a series expansion (see Deming’s remarks in Bayes [1764] 1963). The second article is the famous “An Essay Towards Solving a Problem in the Doctrine of Chances,” with Price’s preface, footnotes, and appendix (followed, a year later, by a continuation and further development of some of Bayes’ results).

The “Problem” posed in the Essay is: *“Given* the number of times in which an unknown event has happened and failed: *Required* the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.” A few sentences later Bayes writes: “By *chance* I mean the same as probability” ([1764] 1963, p. 376).

If the number of successful happenings of the event is *p* and the failures *q,* and if the two named “degrees” of probability are *b* and *f*, respectively, Proposition 9 of the Essay provides the following answer expressed in terms of areas under the curve x^{p}(1 - x)^{q}:

This is based on the assumption (Bayes’ “Postulate 1”) that all values of the unknown probability are equally likely before the observations are made. Bayes indicated the applicability of this postulate in his famous “Scholium”: “that the … rule is the proper one to be used in the case of an event concerning the probability of which we absolutely know nothing antecedently to any trials made concerning it, seems to appear from the following consideration; viz. that concerning such an event I have no reason to think that, in a certain number of trials, it should rather happen any one possible number of times than another” *(ibid.,* pp. 392–393).

The remainder of Bayes’ Essay and the supplement (half of which was written by Price) consists of attempts to evaluate (1) numerically, (*a*) by expansion of the integrand and *(b)* by integration by parts. The results are satisfactory for *p* and *q* small but the approximations for large *p, q* are only of historical interest (Wishart 1927).

Opinions about the intellectual and mathematical ability evidenced by the letter and the essay are extraordinarily diverse. Netto (1908), after outlining Bayes’ geometrical proof, agreed with Laplace ([1812] 1820) that it is *ein wenig verwickelt* (“somewhat involved”). Todhunter (1865) thought that the résumé of probability theory that precedes Proposition 9 was “excessively obscure.” Molina (in Bayes [1764] 1963, p. xi) said that “Bayes and Price … can hardly be classed with the great mathematicians that immediately preceded or followed them,” and Hogben (1957, p. 133) stated that “the ideas commonly identified with the name of Bayes are largely [Laplace’s].”

On the other hand von Wright (1951, p. 292) found Bayes’ Essay “a masterpiece of mathematical elegance and free from … obscure philosophical pretentions.” Barnard (1958, p. 295) wrote that Bayes’ “mathematical work … is of the very highest quality.” Fisher ([1956] 1959, p, 8) concurred with these views when he said Bayes’ “mathematical contributions … show him to have been in the first rank of independent thinkers. …”

The subsequent history of mathematicians’ and philosophers’ extensions and criticisms of Proposition 9—the only statement that can properly be called Bayes’ theorem (or rule)—is entertaining and instructive. In his first published article on probability theory, Laplace (1774), without mentioning Bayes, introduced the principle that if *P _{j}* is the probability of an observable event resulting from “cause”

*j (j =*1, 2, 3, …,

*n*) then the probability that “cause”

*j*is operative to produce the observed event is

This is Principle III of the first (1812) edition of Laplace’s probability text, and it implies that the prior (antecedent, initial) probabilities of each of the “causes” is the same. However, in the second (1814) edition Laplace added a few lines saying that if the “causes” are not equally probable a priori (2) would become

where ω_{j} is the prior probability of cause *j* and p_{j} is now the probability of the event, given that “cause” *j* is operative. He gave no illustrations of this more general formula.

Laplace (1774) applied his new principle (2) to find the probability of drawing *m* white and *n* black tickets in a specified order from an urn containing an infinite number of white and black tickets in an unknown ratio and from which *p* white and *q* black tickets have already been drawn. His solution, namely,

was later (1778–1781; 1812, chapter 6) generalized by the bare statement that if all values of *x* are not equally probable a factor *z(x)* representing the a priori probability density *(facilité)* of *x* must appear in both integrands. However, Laplace’s own views on the applicability of expressions like (4) were stated in 1778 (1778–1781, p. 264) and agree with those of Bayes’ Scholium: “Lorsqu’on n’a aucune donnée *a priori* sur la possibilité d’un événement, il faut supposer toutes les possibilityés, depuiszéro jusqu’àl’unité, également probables. …” (“When nothing is given a priori as to the prob-ability of an event, one must suppose all probabilities, from zero to one, to be equally likely. …”) Much later Karl Pearson (1924, p. 191) pointed out that Bayes was “considering excess of one variate … over a second … as the determining factor of occurrence” and this led naturally to a generalization of the measure in the integrals of (1). Fisher (1956) has even suggested that Bayes himself had this possibility in mind.

Laplace’s views about prior probability distributions found qualified acceptance on the Continent (von Kries 1886) but were subjected to strong criticism in England (Boole 1854; Venn 1866; Chrystal 1891; Fisher 1922), where a relative frequency definition of probability was proposed and found incompatible with the uniform prior distribution (for example, E. S. Pearson 1925). However, developments in the theory of inference (Keynes 1921; Ramsey 1923–1928; Jeffreys 1931; de Finetti 1937; Savage 1954; Good 1965) suggest that there are advantages to be gained from a “subjective” or a “logical” definition of probability and this approach gives Bayes’ theorem, in its more general form, a central place in inductive procedures (Jeffreys 1939; Raiffa & Schlaifer 1961; Lindley 1965).

Hilary L. Seal

*[For the historical context of Eayes’ work, see*Statistics, *article on*the history of statistical method; *and the biography of*Laplace. For *discussion of the subsequent development of his ideas, see*Bayesian inference; Probability; *and the biographies of*Fisher, R. A.; *and*Pearson.]

## BIBLIOGRAPHY

Barnard, G. A. 1958 Thomas Bayes: A Biographical Note. *Biometrika* 45:293–295.

Bayes, Thomas (1764) 1963 *Facsimiles of Two Papers by Bayes.* New York: Hafner. → Contains “An Essay Towards Solving a Problem in the Doctrine of Chances, With Richard Price’s Foreword and Discussion,” with a commentary by Edward C. Molina; and “A Letter on Asymptotic Series From Bayes to John Canton,” with a commentary by W. Edwards Deming. Both essays first appeared in Volume 53 of the *Philosophical Transactions,* Royal Society of London, and retain the original pagination.

Boole, George (1854) 1951 *An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities.* New York: Dover.

Chrystal, George 1891 On Some Fundamental Principles in the Theory of Probability. Actuarial Society of Edinburgh, *Transactions* 2:419–439.

de Finetti, Bruno 1937 La prévision: Ses lois logiques, ses sources subjectives. Paris, Université de, Institut Henri Poincaré, *Annales* 7:1–68.

Fisher, R. A. (1922) 1950 On the Mathematical Foundations of Theoretical Statistics. Pages 10.307a–10.368 in R. A. Fisher, *Contributions to Mathematical Statistics.* New York: Wiley. → First published in Volume 222 of the *Philosophical Transactions,* Series A, Royal Society of London.

Fisher, R. A. (1956) 1959 *Statistical Methods and Scientific Inference.* 2d ed., rev. New York: Hafner; London: Oliver & Boyd.

Good, Irving J. 1965 *The Estimation of Probabilities: An Essay on Modern Bayesian Methods.* Cambridge, Mass.: M.I.T. Press.

Hogben, Lancelot T. 1957 *Statistical Theory; the Relationship of Probability, Credibility and Error: An Examination of the Contemporary Crisis in Statistical Theory From a Behaviourist Viewpoint.* London: Allen & Unwin.

Holland, J. D. 1962 The Reverend Thomas Bayes, F.R.S. (1702–1761). *Journal of the Royal Statistical Society* Series A 125:451–461.

Jeffreys, Harold (1931) 1957 *Scientific Inference.* 2d ed. Cambridge Univ. Press.

Jeffreys, Harold (1939) 1961 *Theory of Probability.* 3d ed. Oxford: Clarendon.

Keynes, J. M. (1921) 1952 *A Treatise on Probability.* London: Macmillan. → A paperback edition was published in 1962 by Harper.

Kries, Johannes von (1886) 1927 *Die Principien der Wahrscheinlichkeitsrechnung: Eine logische Untersuchung.* 2d ed. Tübingen (Germany): Mohr.

Laplace, Pierre S. (1774) 1891 Mémoire sur la probabilité des causes par les événements. Volume 8, pages 27–65 in Pierre S. Laplace, *Oeuvres complétes de Laplace.* Paris: Gauthier-Villars.

Laplace, Pierre S. (1778–1781) 1893 Mémoire sur les probabilités. Volume 9, pages 383–485 in Pierre S. Laplace, *Oeuvres complètes de Laplace.* Paris: Gauthier-Villars.

Laplace, Pierre S. (1812) 1820 *Théorie analytique des probabilités.* 3d ed., rev. Paris: Courcier.

Lindley, Dennis V. 1965 *Introduction to Probability and Statistics From a Bayesian Viewpoint.* 2 vols. Cambridge Univ. Press.

Netto, E. 1908 Kombinatorik, Wahrscheinlichkeitsrechnung, Reihen-Imaginäres. Volume 4, pages 199–318, in Moritz Cantor (editor), *Vorlesungen über Geschichte der Mathematik.* Leipzig: Teubner.

Ogborn, Maurice E. 1962 *Equitable Assurances: The Story of Life Assurance in the Experience of The Equitable Life Assurance Society, 1762–1962.* London: Allen & Unwin.

Pearson, Egon S. 1925 Bayes’ Theorem, Examined in the Light of Experimental Sampling. *Biometrika* 17: 388–442.

Pearson, Karl 1924 Note on Bayes’ Theorem. *Biometrika* 16:190–193.

Price, Richard 1765 A Demonstration of the Second Rule in the Essay Towards a Solution of a Problem in the Doctrine of Chances. Royal Society of London, *Philosophical Transactions* 54:296–325. → Reprinted by Johnson in 1965.

Raiffa, Howard; and Schlaifer, Robert 1961 *Applied Statistical Decision Theory.* Harvard University Graduate School of Business Administration, Studies in Managerial Economics. Boston: The School.

Ramsey, Frank P. (1923–1928) 1950 *The Foundations of Mathematics and Other Logical Essays.* New York: Humanities.

Savage, Leonard J. 1954 *The Foundations of Statistics.* New York: Wiley.

Todhunter, Isaac (1865) 1949 A *History of the Mathematical Theory of Probability From the Time of Pascal to That of Laplace.* New York: Chelsea.

Venn, John (1866) 1888 *The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability, With Special Reference to Its Logical Bearings and Its Application to Moral and Social Science.* 3d ed. London: Macmillan.

Wishart, John 1927 On the Approximate Quadrature of Certain Skew Curves, With an Account of the Re-searches of Thomas Bayes. *Biometrika* 19:1–38.

Wright, Georg H. von 1951 *A Treatise on Induction and Probability.* London: Routledge.

# Bayes, Thomas

# Bayes, Thomas

(*b*. London, England, 1702; *d*. Tunbridge Wells, England, 17 April 1761)

*probability.*

Bayes was a member of the first secure generation of English religious Nonconformists. His father, Joshua Bayes, F.R.S., was a respected theologian of dissent; he was also one of the group of six ministers who were the first to be publicly ordained as Nonconformists. Privately educated, Bayes became his father’s assistant at the presbytery in Holborn, London; his mature life was spent as minister at the chapel in Tunbridge Wells. Despite his provincial circumstances, he was a wealthy bachelor with many friends. The Royal Society of London elected him a fellow in 1742. He wrote little: *Divine Benevolence* (1731) and *Introduction to the Doctrine of Fluxions* (1736) are the only works known to have been published during his lifetime. The latter is a response to Bishop Berkeley’s *Analyst*, a stinging attack on the logical foundations of Newton’s calculus; Bayes’s reply was perhaps the soundest retort to Berkeley then available.

Bayes is remembered for his brief “Essay Towards Solving a Problem in the Doctrine of Chances” (1763), the first attempt to establish foundations for statistical inference. Jacques Bernoulli’s *Ars Conjectandi* (1713) and Abraham de Moivre’s *The Doctrine of chances* (1718) already provided great textbooks of what we now call probability theory. Given the probability of one event, the logical principles for inferring the probabilities of related events were quite well understood. In his “Essay,” Bayes set himself the “converse problem”: “*Given* the number of times in which an unknown event has happened and failed: *Required* the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.” “By chance,” he said, “I mean the same as probability.”

In the light of Bernoulli’s *Ars conjectandi*, and of a paper by John Arbuthnot (1710), there was some understanding of how to reject statistical hypotheses in the light of data; but no one had shown how to measure the probability of statistical hypotheses in the light of data. Bayes began his solution of the problem by noting that sometimes the probability of a statistical hypothesis is given before any particular events are observed; he then showed how to compute the probability of the hypothesis after some observations are made. In his own example:

Postulate: 1. I suppose the square table or plane ABCD to be so made and levelled that if either of the balls O or W be thrown upon it, there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it.

2. I suppose that the ball W shall be first thrown, and through the point where it rests a line

osshall be drawn parallel to AD, and meeting CD and AB insando, and that afterwards the ball O shall be thrownp+qorntimes, and that its resting between AD andosafter a single throw be called the happening of the event M in a single trial.

For any fractions *f* and *b* (between zero and one), Bayes was concerned with the probability of assertions of the form

f≤ probability of M ≤b.

From his physical assumptions about the table ABCD, he inferred that the prior probability (i.e., the probability before any trials have been made) is *b* — *f*. He proved, as a theorem in direct probabilities, that the posterior probability (i.e., the probability, on the evidence, that M occurred *p* times and failed *q* times) is

A generalization on this deduction is often, anachronistically, called Bayes’s theorem or Bayes’s formula. In the case of only finitely many statistical hypotheses, *H*_{1},..., *H*_{n}, let there be prior data *D* and some new observed evidence *E*. Then the prior probability of *H*_{i} is Prob (*H*_{i}/*D*); the posterior probability is Prob (*H*_{i}/*D* and *E*); the theorem asserts:

Prob (*H*_{i}/*D* and *E*)

A corresponding theorem holds in the continuum; Bayes’s deduction essentially involves a special case of it.

The work so far described falls entirely within probability theory, and would now be regarded as a straightforward deduction from standard probability axioms. The striking feature of Bayes’s work is an argument found in a *scholium* to the paper, which does not follow from any standard axioms. Suppose we have no information about the prior probability of a statistical hypothesis. Bayes argues by analogy that, in this case, our ignorance is neither more nor less than in his example where prior probabilities are known to be entirely uniform. He concludes, “I shall take for granted that the rule given concerning the event M... is also the rule to be used in relation to any event concerning the probability of which nothing at all is known antecedently to any trials made or observed concerning it.”

If Bayes’s conclusion is correct, we have a basis for the whole of statistical inference. Richard Price, who sent Bayes’s paper to the Royal Society, seems to imply in a covering letter that Bayes was not satisfied with his argument by analogy and, hence, had declined to publish it. Whatever the case with Bayes, Laplace had no qualms about Bayes’s argument; and from 1774 he regularly assumed uniform prior probability distributions. His enormous influence made Bayes’s ideas almost unchallengeable until George Boole protested in his *Laws of Thought* (1854). Since then, Bayes’s technique has been a constant subject of controversy.

Today there are two kinds of Bayesians. Sir Harold Jeffreys, in his *Theory of Probability*, maintains that, relative to any body of information, even virtual ignorance, there is an objective distribution of degrees of confidence appropriate to various hypotheses; he often rejects Bayes’s actual postulate, but accepts the need for similar postulates. Leonard J. Savage, in his *Foundations of Statistics*, rejects objective probabilities, but interprets probability in a personal way, as reflecting a person’s personal degree of belief; hence, a prior probability is a person’s belief before he has made some observations, and his posterior probability is his belief after the observations are made. Many working statisticians who are Bayesians, in the sense of trying to argue from prior probabilities, try to be neutral between Jeffreys and Savage. In this respect they are perhaps close to Bayes himself. He defined the probability of an event as “the ratio between the value at which an expectation depending upon the happening of the event ought to be computed, and the value of the thing expected upon its happening.” This definition can be interpreted in either a subjective or an objective way, but there is no evidence that Bayes had even reflected on which interpretation he might prefer.

## BIBLIOGRAPHY

I. Original Works. Bayes’s works published during his lifetime are *Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government Is the Happiness of His Creatures* (London, 1731); and *An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of* The Analyst (London, 1736). “An Essay Towards Solving a Problem in the Doctrine of Chances” was published in *Philosophical Transactions of the Royal Society of London*, **53** (1763), 370–418, with a covering letter written by Richard Price; repr. in *Biometrika*, **45** (1958), 296–315, with a biographical note by G. A. Barnard. Also of interest is “A Letter on Asymptotic Series from Bayes to John Canton,” in *Biometrika*, **45** (1958), 269–271; repr. with the paper from the *Philosophical Transactions in Facsimiles of Two Papers by Bayes* (Washington, D.C., n.d.), with a commentary on the first by Edward C. Molina and on the second by W. Edwards Deming.

II. Secondary Literature. Supplementary information may be found in John Arbuthnot, “An Argument for Divine Providence Taken From the Constant Regularity of the Births of Both Sexes,” in *Philosophical Transactions of the Royal Society of London*, **23** (1710), 186–190; George Berkeley, *The Analyst* (London-Dublin, 1734); Jacques Bernoulli, *Ars conjectandi* (Basel, 1713); George Boole, *An Investigation of the Laws of Thought* (London, 1854); Harold Jeffreys, *Theory of Probability* (Oxford, 1939; 3rd ed., 1961); Pierre Simon Laplace, “Mémoire sur la probabilité des causes par les événements,” in *Mémoires par divers savants*, **6** (1774), 621–656, and *Théorie analytique des probabilités* (Paris, 1812); Abraham de Moivre, *The Doctrine of Changes* (London, 1718; 3rd ed., 1756); and Leonard J. Savage, *The Foundations of Statistics* (New York-London, 1954).

Ian Hacking

# Bayes, Thomas

# BAYES, THOMAS

(*b*. Bovingdon, Hertfordshire[?], England, 1702; *d*. Tunbridge Wells, Kent, England, 7 April 1761)

*probability*. For the original article on Bayes see *DSB*, vol. 1.

Bayes is rightly remembered chiefly for his work on the determination of the support for a statistical hypothesis provided by experimental data (or, on the determination of the probability of a cause from knowledge of an observed occurrence). However, he also provided results concerning the approximation of the skew beta distribution, the series expansion of log *z*!, and (with Richard Price) a series expansion of the Normal probability integral.

**Genealogy.** The eldest of the seven children of Joshua (who was *not* a Fellow of the Royal Society) and Anne (née Carpenter) Bayes, Thomas might well have been born in Bovingdon, near Hemel Hempstead in Hertfordshire. William Urwick (1884, p. 390) states that Joshua Bayes ministered at the Box Lane Chapel, in Bovingdon, after his ordination (in 1694), and remained there for about eleven years before returning to London. There he became an assistant to John Sheffield at St. Thomas's, Southwark, and he subsequently officiated at the dissenting chapel in Leather Lane in Hatton Gardens. Here Thomas later assisted him, before moving to Tunbridge Wells in the early 1730s.

**Education.** While it is uncertain where Thomas received his early education, there is no doubt that he later studied at Edinburgh University. The records of that institution for 1719 (Scottish style) show that Thomas was admitted on the nineteenth of February, and his name is also recorded in the *List of Theologues in the College of Edinburgh since October 1711*. Like many intending to become Nonconformist ministers, he did not take the MA degree, and returned to London licensed but not ordained.

The certificate detailing Bayes’s admission to the university library was signed by James Gregory, brother of the Scottish mathematician and astronomer David Gregory, his immediate predecessor in the chair of mathematics, and nephew of the illustrious James Gregory (1638–1675). David Gregory has been credited with the introduction of Isaac Newton’s *Principia* to Edinburgh students, and it is possible that his brother James continued with this practice. Thus Bayes might have been exposed to the fluxionary calculus, and this could well have accounted for the precision and understanding evinced in his *Introduction to the Doctrine of Fluxions*.

**Works.** In his *Divine Benevolence* Bayes argued that the works of God are inspired and motivated by benevolence, contra the view expressed in John Balguy’s earlier tract that rectitude was the guiding principle. Subsequently Henry Grove saw wisdom as God’s most fundamental attribute. All three of these tracts were studied at dissenting academies of that period.

In addition to *An Essay towards Solving a Problem in the Doctrine of Chances*, volume 53 of the *Philosophical Transactions of the Royal Society* contains a *Letter* from Bayes on a semiconvergent series. Here he showed that the well-known James Stirling–Abraham de Moivre series “for” log *z*! in fact diverges for any *z* (the divergence for *z* = 1 had been shown by the Swiss mathematician Leon-hard Euler some six years before Bayes’s death).

On the matter of the scholium in the *Essay*, Stephen M. Stigler interprets Bayes as showing that, for the balls-and-table example, *Prob(M=p)=1/(n+1)* for *p=0, 1, …, n*, a characterization of the uniform distribution on *[0,1]* due to Bayes. Suppose next that we have an event (say *E*) of whose probability (say 0) we are ignorant. Because the probability that *E* occurs *p* times in *n* trials is also *1/(n+1)*, 0 must again be uniformly distributed. That is, that 0 has a uniform distribution is implied by the constancy of *Prob(E=p)*. When the scholium is viewed in this way, the problems sometimes thought to arise from the apparently arbitrary choice of a uniform prior disappear.

The *Essay* was followed by a *Supplement*, partly by Richard Price, published in volume 54 of the *Philosophical Transactions*. In addition to Bayes’s famous result, these two papers contain, in the rules, an approximation to the two-sided beta probability integral that is considerably better than the Normal approximation (in essence, the skew beta probability density function, obtained by Bayes as a posterior density, is approximated by a symmetric beta probability function multiplied by a factor tending in the limit to 1, the two betas having the same maximum and points of inflexion). Also to be found in the *Supplement* is a series expansion, by Price, of the Normal approximation to the posterior distribution derived in the *Essay*.

In the library of the Royal Society of London is a letter from Bayes to John Canton commenting on work by Thomas Simpson on errors in observations. Bayes’s view was that an increase in the number of observations will not cause a decrease in the probability of error if the measuring device is inaccurate. With this is connected Simpson’s assumption that errors in excess and in defect of the true value are equally probable.

A serendipitous find by David R. Bellhouse in the Kent County Archives in Maidstone, U.K., suggests that Bayes might well have been used by Philip Stanhope, 2nd Earl Stanhope, as a mathematical critic and commentator. Bellhouse discovered several items in the Stanhope Collection: (1) preliminary drafts of Bayes’s paper on series; (2) a derivation, in Bayes’s hand, of the series for log *z*! that does not use the semiconvergent series; and (3) letters between Bayes and Stanhope, supportive of the above suggestion, referring to some observations by Patrick Murdoch. There is also a note, in Stanhope’s writing, headed “The Reverend Mr Bayes’s Paper concerning Trinomial divisors.”

The Equitable Life Assurance Society in London holds a notebook by Bayes. With passages in French, Latin, English, and an obscure seventeenth-century shorthand, the notebook contains notes on matters as diverse as mathematics, electricity, optics, and music.

**Death.** Although giving up his ministry in 1752, Bayes probably remained in Tunbridge Wells, where he died in 1761, being interred in the family vault in the Bunhill Fields Burial Ground, near Moorgate, London. Despite frequent restorations of the vault, the inscriptions have at times been difficult to decipher. This is perhaps the reason for the Canadian philosopher Ian Hacking’s recording of Thomas’s death as the seventeenth, rather than the seventh, of April.

## SUPPLEMENTARY BIBLIOGRAPHY

Bellhouse, David R. “On Some Recently Discovered Manuscripts of Thomas Bayes.” *Historia Mathematica* 29, no. 4 (2002): 383–394.

———. “The Reverend Thomas Bayes, F.R.S.: A Biography to Celebrate the Tercentenary of His Birth.” *Statistical Science*19, no. 1 (2004): 3–43.

Dale, Andrew I. *Most Honourable Remembrance: The Life and Work of Thomas Bayes*. New York: Springer-Verlag, 2003. Reprints of and commentaries on Bayes’s works.

Grant, Alexander. *The Story of the University of Edinburgh during Its First Three Hundred Years*. 2 vols. London: Longmans, Green, 1884.

Hald, Anders. *A History of Mathematical Statistics from 1750 to 1930*. New York: Wiley, 1998.

Stigler, Stephen M. “Thomas Bayes’s Bayesian Inference.’’ *Journal of the Royal Statistical Society*, ser. A, 145, pt. 2 (1982): 250–258.

Urwick, William. *Nonconformity in Herts: Being Lectures upon the Nonconforming Worthies of St. Albans, and Memorials of Puritanism and Nonconformity in all the Parishes of the County of Hertford*. London: Hazell, Watson, and Viney, 1884.

*Andrew I. Dale*

# Bayes, Thomas

# BAYES, THOMAS

An English country clergyman, amateur mathematician, and inveterate gambler, Thomas Bayes (1702–1761) is remembered for his development of ideas and concepts in the theory of probability. These are described in his *Essay Towards Solving a Problem in the Doctrine of Chances*, published posthumously in 1763. Bayes was interested in the chances of drawing a winning hand in card games, throwing the right combination of numbers with a pair of dice, and picking the winner in a horse race. He expounded on the chance of events occurring on the basis of preexisting circumstances and after the occurrence of particular events, which he termed "prior odds" (or probability) and "posterior odds." His *Essay* was rediscovered in the twentieth century and was put to service in Bayesian statistics, a branch of stochastic mathematics that does not use statistical significance tests. It has proved very useful in decision analysis, clinical epidemiology, health-services research, and other applications of probability theory.

John M. Last

(see also *Bayes' Theorem; Probability Model; Statistics for Public Health* )

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