probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, inverse probability is an obsolete term for the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

of an unobserved variable. Today, the problem of determining an unobserved variable (by whatever method) is called

inferential statistics Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

, the method of inverse probability (assigning a probability distribution to an unobserved variable) is called

Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...

, the "distribution" of data given the unobserved variable is rather the

likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...

(which is not a probability distribution), and the distribution of an unobserved variable, given both data and a

prior distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...

, is the

posterior distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...

. The development of the field and terminology from "inverse probability" to "Bayesian probability" is described by . Youngronaldfisher2

The term "inverse probability" appears in an 1837 paper of De Morgan, in reference to

Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

's method of probability (developed in a 1774 paper, which independently discovered and popularized Bayesian methods, and a 1812 book), though the term "inverse probability" does not occur in these. Fisher uses the term in , referring to "the fundamental paradox of inverse probability" as the source of the confusion between statistical terms that refer to the true value to be estimated, with the actual value arrived at by the estimation method, which is subject to error. Later Jeffreys uses the term in his defense of the methods of Bayes and Laplace, in . The term "Bayesian", which displaced "inverse probability", was introduced by

Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...

in 1950. Inverse probability, variously interpreted, was the dominant approach to statistics until the development of frequentism in the early 20th century by

Jerzy Neyman Jerzy Neyman (April 16, 1894 – August 5, 1981; born Jerzy Spława-Neyman; ) was a Polish mathematician and statistician who spent the first part of his professional career at various institutions in Warsaw, Poland and then at University Colleg ...

and

Egon Pearson Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a leading British statistician. Career He was educated at Winchester College and Trinity College ...

. Following the development of frequentism, the terms

frequentist Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or pro ...

and

Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...

developed to contrast these approaches, and became common in the 1950s.

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In modern terms, given a probability distribution ''p''(''x'', θ) for an observable quantity ''x'' conditional on an unobserved variable θ, the "inverse probability" is the

''p''(θ, ''x''), which depends both on the likelihood function (the inversion of the probability distribution) and a prior distribution. The distribution ''p''(''x'', θ) itself is called the direct probability. The ''inverse probability problem'' (in the 18th and 19th centuries) was the problem of estimating a parameter from experimental data in the experimental sciences, especially

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

and

biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...

. A simple example would be the problem of estimating the position of a star in the sky (at a certain time on a certain date) for purposes of

navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation ...

. Given the data, one must estimate the true position (probably by averaging). This problem would now be considered one of

. The terms "direct probability" and "inverse probability" were in use until the middle part of the 20th century, when the terms "

" and "posterior distribution" became prevalent.

References

* ** See reprint in * * {{DEFAULTSORT:Inverse Probability Statistical inference Probability interpretations Bayesian statistics

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See also

References