Range theories of probability (19TH CENTURY)

Developed by French mathematician Laplace (1749-1827) Certain theories analyzing probability in terms of ranges of alternatives.

William Calvert Kneale (1906-1990) introduces such a theory to deal with paradoxes that face the classical theory of probability when the relevant range of alternatives is infinite, and his theory consists basically of an extension to the classical theory to cover those cases.

Source:
W C Kneale, Probability and Induction (1949), 35

Interpretations

When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. For example, tossing a fair coin twice will yield “head-head”, “head-tail”, “tail-head”, and “tail-tail” outcomes. The probability of getting an outcome of “head-head” is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:

1. Objectivists assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment’s outcome, when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency “in the long run” of outcomes.^[4] A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once.
2. Subjectivists assign numbers per subjective probability, that is, as a degree of belief.^[5] The degree of belief has been interpreted as “the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.”^[6] The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date.^[7] By Aumann’s agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.^[8]

Etymology

The word probability derives from the Latin probabilitas, which can also mean “probity”, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness’s nobility. In a sense, this differs much from the modern meaning of probability, which in contrast is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.^[9]

History

The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues^{[clarification needed]} are still obscured by the superstitions of gamblers.^[10]

According to Richard Jeffrey, “Before the middle of the seventeenth century, the term ‘probable’ (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances.”^[11] However, in legal contexts especially, ‘probable’ could also apply to propositions for which there was good evidence.^[12]

Al-Kindi’s Book of Cryptographic Messages contains the earliest known use of statistical inference (9th century)

The earliest known forms of probability and statistics were developed by Middle Eastern mathematicians studying cryptography between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. Al-Kindi (801–873) made the earliest known use of statistical inference in his work on cryptanalysis and frequency analysis. An important contribution of Ibn Adlan (1187–1268) was on sample size for use of frequency analysis.^[13]

Gerolamo Cardano (16th century)

Christiaan Huygens published one of the first books on probability (17th century)

The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes^[14]). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.^[15] Jakob Bernoulli’s Ars Conjectandi (posthumous, 1713) and Abraham de Moivre’s Doctrine of Chances (1718) treated the subject as a branch of mathematics.^[16] See Ian Hacking’s The Emergence of Probability^[9] and James Franklin’s The Science of Conjecture^[17] for histories of the early development of the very concept of mathematical probability.

The theory of errors may be traced back to Roger Cotes’s Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.^[18] The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

The first two laws of error that were proposed both originated with Pierre-Simon Laplace. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error—disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error.^[19] The second law of error is called the normal distribution or the Gauss law. “It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old.”^[19]

Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

Carl Friedrich Gauss

Adrien-Marie Legendre (1805) developed the method of least squares, and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).^[20] In ignorance of Legendre’s contribution, an Irish-American writer, Robert Adrain, editor of “The Analyst” (1808), first deduced the law of facility of error,

{\displaystyle \phi (x)=ce^{-h^{2}x^{2}},}

where {\displaystyle h} is a constant depending on precision of observation, and {\displaystyle c} is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel’s (1850).^{[citation needed]} Gauss gave the first proof that seems to have been known in Europe (the third after Adrain’s) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters’s (1856) formula^{[clarification needed]} for r, the probable error of a single observation, is well known.

In the nineteenth century, authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

In 1906, Andrey Markov introduced^[21] the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov in 1931.^[22]

On the geometric side, contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).^[23] See integral geometry for more info.