Thomas Bayes’s work on the inverse problem in probabilities, which attempted to calculate the probabilities of causes from those of events, helped to advance investigations in the foundations of probability. Bayes’s theorem is a major part of subjectivist approaches to epistemology, statistics, and inductive logic.
Thomas Bayes was a clergyman known for his mathematical interests. Little else is known of his life, however, despite extensive research by scholars interested in his work. He was educated at the University of Edinburgh in Scotland, and he spent most of his life as a minister in Tunbridge Wells, a fashionable spa not far from London. He became a fellow of the Royal Society of London in 1742, and after his death in 1761, his manuscripts came to the attention of the Reverend Richard Price, who communicated Bayes’s essay about probability to the Royal Society. The essay, “An Essay Toward Solving a Problem in the Doctrine of Chances,”
Bayes was deeply interested in confronting some of the fundamental issues in probability. Even though some of his views have not found universal acceptance, some of the results given in his essay have proved to be both important technical tools in probability and means for understanding the foundations of the subject. The essay itself did not long retain the interests of the scientific community, but its results were brought back to the attention of those interested in probability in the nineteenth century, and his ideas have remained central in this area of mathematics.
There had been quite a tradition of probability texts by the time that Bayes began writing, the most prominent (in English) being that of Abraham de Moivre (The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play,
Bayes started off with a definition of probability that differed from that of de Moivre, to whose work he makes no explicit reference. The tradition had been to regard the probability of an event as a fraction, which worked out if the sample space (the collection of possible outcomes) was discrete. If the sample space were continuous, however, then a definition using calculus might have been expected. Bayes avoids using explicit notions from calculus by referring to geometrical areas instead. One of the few publications of Bayes that came out during his lifetime was a defense of the ideas of the calculus as laid out by Sir Isaac Newton, so it is clear that he was well acquainted with the state of the calculus. It is unclear whether his move from calculus in his presentation of his ideas about probability was motivated by the wish for a wider readership or in deference to the geometrical style in which Newton had himself written his Philosophiae naturalis principia mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as the Principia, 1848). Bayes seems to have been determined in being excruciatingly careful in justifying his steps, and he may have felt that the geometrical arguments lent themselves to more thorough scrutiny than arguments from calculus.
The problem that Bernoulli had originally confronted had to do with statistical inference in a binomial distribution. Given a certain number of successes out of a certain number of trials, the determination of the probability of a success on a single trial presumably involved some sort of algebraic combination of the numbers. The difficulty Bernoulli faced, however, was that the values that emerged were too complicated to calculate. Bayes gave himself the advantage of dealing with a continuous distribution (represented by points in the plane), which got around the algebraic difficulties that made Bernoulli’s task insurmountable.
The image given by Bayes as the starting point for his calculations in his essay was that of a table on which balls were rolled. The table was marked off in different regions, and the locations where the balls stopped were recorded. On the basis of this model, Bayes was able to calculate what the probability was that a given ball would stop at a given place. He does not provide a justification for the choice of model, and there was a good deal of criticism in subsequent centuries over whether that picture does represent the kind of probability necessary for analyzing more general situations.
In particular, Bayes seeks to justify his model by working on the assumption that if there is no information about the distribution of the results of an experiment, there should be an equal chance of its taking on any of the possible outcomes. While Bayes is fairly careful in his statement of the circumstances under which this ignorance is allowed to dictate the distribution, most subsequent readers of his text looked at it through the lens of the later “principle of insufficient reason” introduced by the French mathematician Pierre-Simon Laplace.
Bayes also indicates in his essay how to calculate some of the areas that arise. This is a tribute to his strength as a mathematician, and the numbers serve to illustrate the reasonableness of his approach to the inverse problem. It is unclear how much editing was done by Price in sending the manuscript to the Royal Society, although he certainly replaced Bayes’s original introduction with his own. Some of Bayes’s points might be easier to understand if his introduction had survived.
There is a result in probability theory known as Bayes’s theorem. The theorem gives a value for the probability of one event (given another) in terms of prior probabilities and the probability of the second event given the first. It is a relatively simple formula in the case of a discrete distribution and does not appear explicitly in Bayes’s essay. Nevertheless, the use of his name for the theorem is perhaps justified by the care with which he examined the application of observation to the calculation of probabilities. Although Bayes himself does not refer to the probabilities of “causes,” his technique has been used to examine many issues in the philosophy of science proceeding from observation.
Thomas Bayes’s work did not receive much attention from nineteenth century thinkers and writers on the subject of probability. It is not clear why he received so little attention, except that his lack of status in the field of mathematics during his lifetime may have provided little impetus for the warranted attention. Also, it is not clear why Bayes failed to publish the essay during his lifetime, although it is possible that he would not want it published because he found the philosophical side of the paper unpersuasive. It might be the case that he was continuing to work on the technical aspects of calculating the values for the expressions that involved areas.
Laplace’s approach to the problem of inverse probabilities was, for many years, thought to have settled the issue of the best form such a theory should take. In the twentieth century, however, many mathematicians and philosophers began to look at Laplace’s arguments more carefully and to examine Bayes’s approach afresh. This led to Bayesianism emerging as the dominant philosophical approach to probability, since it could fit into either the objective perspective (which takes probabilities as measurable quantities in the world) or the subjective view (which takes probabilities as reflections of beliefs and attitudes). Some philosophers of probability have even tried to go back further into the era before Bayes, but the majority of probabilists continue to find Bayes’s approach, as presented in his essay, helpful and persuasive.
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Bayes, Thomas. Facsimiles of Two Papers by Bayes. I. “An Essay Toward Solving a Problem in the Doctrine of Chances,” and II. “A Letter on Asymptotic Series from Bayes to John Canton.” New York: Hafner, 1963. A reprint of the 1940 facsimile edition of Bayes’s major paper and an additional article from the same journal volume of 1763. The first article includes Richard Price’s foreword and his discussion of Bayes’s essay. -
Dale, Andrew I. A History of Inverse Probability. New York: Springer-Verlag, 1991. Follows the reception of Bayes’s work to the beginning of the twentieth century. -
_______. Most Honourable Remembrance: The Life and Work of Thomas Bayes. New York: Springer, 2003. The most complete biography, which includes Bayes’s essay. -
Hald, Anders. A History of Mathematical Statistics from 1750 to 1930. New York: John Wiley and Sons, 1998. A thorough examination of the technical parts of Bayes’s essay. -
Lindley, D. V. “Thomas Bayes.” In Statisticians of the Centuries. New York: Springer, 2001. Connects Bayes’s work to the twentieth century. -
Shafer, Glenn. “Bayes’s Two Arguments for the Rule of Conditioning.” Annals of Statistics 10 (1982): 1075-1089. A scrupulous, critical reading of Bayes’s text. -
Stigler, Stephen M. The History of Statistics. Cambridge, Mass.: Harvard University Press, 1986. Perhaps the best explanation of Bayes’s essay. -
Todhunter, I. A History of the Mathematical Theory of Probability. New York: Chelsea, 1949. Reconstructs Bayes’s mathematical arguments.
Bernoulli Publishes His Calculus of Variations
De Moivre Describes the Bell-Shaped Curve
Maclaurin’s Gravitational Theory
D’Alembert Develops His Axioms of Motion
Agnesi Publishes Analytical Institutions
Euler Develops the Concept of Function
Legendre Introduces Polynomials
Maria Gaetana Agnesi; Jean le Rond d’Alembert; George Berkeley; Leonhard Euler; Joseph-Louis Lagrange; Colin Maclaurin; Gaspard Monge.