Statistics, in the modern sense of the word, began evolving in the 18th century in response to the novel needs of industrializingsovereign states.

In early times, the meaning was restricted to information about states, particularlydemographics such as population. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as innational accounts andtemperature record, and analytical work which requiresstatistical inference. Statistical activities are often associated with models expressed usingprobabilities, hence the connection with probability theory. The large requirements of data processing have made statistics a key application of computing. A number of statistical concepts have an important impact on a wide range of sciences. These include thedesign of experiments and approaches to statistical inference such asBayesian inference, each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics.

By the 18th century, the term "statistics" designated thesystematic collection ofdemographic andeconomic data by states. For at least two millennia, these data were mainly tabulations of human and material resources that might be taxed or put to military use. In the early 19th century, collection intensified, and the meaning of "statistics" broadened to include the discipline concerned with the collection, summary, and analysis of data. Today, data is collected and statistics are computed and widely distributed in government, business, most of the sciences and sports, and even for many pastimes. Electroniccomputers have expedited more elaboratestatistical computation even as they have facilitated the collection and aggregation of data. A single data analyst may have available a set of data-files with millions of records, each with dozens or hundreds of separate measurements. These were collected over time from computer activity (for example, a stock exchange) or from computerized sensors, point-of-sale registers, and so on. Computers then produce simple, accurate summaries, and allow more tedious analyses, such as those that require inverting a large matrix or perform hundreds of steps of iteration, that would never be attempted by hand. Faster computing has allowed statisticians to develop "computer-intensive" methods which may look at all permutations, or use randomization to look at 10,000 permutations of a problem, to estimate answers that are not easy to quantify by theory alone.

The term "mathematical statistics" designates the mathematical theories ofprobability andstatistical inference, which are used instatistical practice. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly usedprobability theory, whose initial results were found in the 17th and 18th centuries, particularly in the analysis ofgames of chance (gambling). By 1800, astronomy used probability models and statistical theories, particularly themethod of least squares. Early probability theory and statistics was systematized in the 19th century and statistical reasoning and probability models were used by social scientists to advance the new sciences ofexperimental psychology andsociology, and by physical scientists inthermodynamics andstatistical mechanics. The development of statistical reasoning was closely associated with the development ofinductive logic and thescientific method, which are concerns that move statisticians away from the narrower area of mathematical statistics. Much of the theoretical work was readily available by the time computers were available to exploit them. By the 1970s,Johnson and Kotz produced a four-volumeCompendium on Statistical Distributions (1st ed., 1969–1972), which is still an invaluable resource.

Applied statistics can be regarded as not a field ofmathematics but an autonomousmathematical science, likecomputer science andoperations research. Unlike mathematics, statistics had its origins inpublic administration. Applications arose early indemography andeconomics; large areas of micro- and macro-economics today are "statistics" with an emphasis on time-series analyses. With its emphasis on learning from data and making best predictions, statistics also has been shaped by areas of academic research including psychological testing, medicine andepidemiology. The ideas of statistical testing have considerable overlap withdecision science. With its concerns with searching and effectively presentingdata, statistics has overlap withinformation science andcomputer science.

Look upstatistics inWiktionary, the free dictionary.

The termstatistics is ultimately derived from theNeo-Latinstatisticum collegium ("council of state") and theItalian wordstatista ("statesman" or "politician"). TheGermanStatistik, first introduced byGottfried Achenwall (1749), originally designated the analysis ofdata about thestate, signifying the "science of state" (then calledpolitical arithmetic in English). It acquired the meaning of the collection and classification of data generally in the early 19th century. It was introduced into English in 1791 bySir John Sinclair when he published the first of 21 volumes titledStatistical Account of Scotland.^[1]

Basic forms of statistics have been used since the beginning of civilization. Early empires often collated censuses of the population or recorded the trade in various commodities. TheHan dynasty and theRoman Empire were some of the first states to extensively gather data on the size of the empire's population, geographical area and wealth.

The use of statistical methods dates back to at least the 5th century BCE. The historianThucydides in hisHistory of the Peloponnesian War^[2] describes how the Athenians calculated the height of the wall ofPlatea by counting the number of bricks in an unplastered section of the wall sufficiently near them to be able to count them. The count was repeated several times by a number of soldiers. The most frequent value (in modern terminology – themode) so determined was taken to be the most likely value of the number of bricks. Multiplying this value by the height of the bricks used in the wall allowed the Athenians to determine the height of the ladders necessary to scale the walls.^[3]

TheTrial of the Pyx is a test of the purity of the coinage of theRoyal Mint which has been held on a regular basis since the 12th century. The Trial itself is based on statistical sampling methods. After minting a series of coins – originally from ten pounds of silver – a single coin was placed in the Pyx – a box inWestminster Abbey. After a given period – now once a year – the coins are removed and weighed. A sample of coins removed from the box are then tested for purity.

TheNuova Cronica, a 14th-centuryhistory of Florence by the Florentine banker and officialGiovanni Villani, includes much statistical information on population, ordinances, commerce and trade, education, and religious facilities and has been described as the first introduction of statistics as a positive element in history,^[4] though neither the term nor the concept of statistics as a specific field yet existed.

The arithmeticmean, although a concept known to the Greeks, was not generalised to more than two values until the 16th century. The invention of the decimal system bySimon Stevin in 1585 seems likely to have facilitated these calculations. This method was first adopted in astronomy byTycho Brahe who was attempting to reduce the errors in his estimates of the locations of various celestial bodies.

The idea of themedian originated inEdward Wright's book on navigation (Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations. The difference between the mean and the median was noticed in 1669 by Chistiaan Huygens in the context of using Graunt's tables.^[5]

The term 'statistic' was introduced by the Italian scholarGirolamo Ghilini in 1589 with reference to this science.^[6][7] The birth of statistics is often dated to 1662, whenJohn Graunt, along withWilliam Petty, developed early human statistical andcensus methods that provided a framework for moderndemography. He produced the firstlife table, giving probabilities of survival to each age. His bookNatural and Political Observations Made upon the Bills of Mortality used analysis of themortality rolls to make the first statistically based estimation of the population ofLondon. He knew that there were around 13,000 funerals per year in London and that three people died per eleven families per year. He estimated from the parish records that the average family size was 8 and calculated that the population of London was about 384,000; this is the first known use of aratio estimator.Laplace in 1802 estimated the population of France with a similar method; seeRatio estimator § History for details.

Although the original scope of statistics was limited to data useful for governance, the approach was extended to many fields of a scientific or commercial nature during the 19th century. The mathematical foundations for the subject heavily drew on the newprobability theory, pioneered in the 16th century byGerolamo Cardano,Pierre de Fermat andBlaise Pascal.Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.Jakob Bernoulli'sArs Conjectandi (posthumous, 1713) andAbraham de Moivre'sThe Doctrine of Chances (1718) treated the subject as a branch of mathematics. In his book Bernoulli introduced the idea of representing complete certainty as one and probability as a number between zero and one.

In 1700,Isaac Newton carried out the earliest known form oflinear regression, writing the first of theordinary least squares normal equations, averaging astronomical data, and summing the residuals to zero in his analysis ofHipparchus’s equinox observations. He distinguished between two inhomogeneous sets of data and might have thought of an optimal solution in terms of bias, but not in effectiveness.^[8][9]

A key early application of statistics in the 18th century was to thehuman sex ratio at birth.^[10]John Arbuthnot studied this question in 1710.^{[11][12][13][14]} Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 0.5^82, or about 1 in 4,8360,0000,0000,0000,0000,0000; in modern terms, thep-value. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." This is and other work by Arbuthnot is credited as "the first use ofsignificance tests"^[15] the first example of reasoning aboutstatistical significance and moral certainty,^[16] and "... perhaps the first published report of anonparametric test ...",^[12] specifically thesign test; see details atSign test § History.

The formal study oftheory of errors may be traced back toRoger Cotes'Opera Miscellanea (posthumous, 1722), but a memoir prepared byThomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down theaxioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given. Simpson discussed several possible distributions of error. He first considered theuniform distribution and then the discrete symmetrictriangular distribution followed by the continuous symmetric triangle distribution.Tobias Mayer, in his study of thelibration of themoon (Kosmographische Nachrichten, Nuremberg, 1750), invented the first formal method for estimating the unknown quantities by generalized the averaging of observations under identical circumstances to the averaging of groups of similar equations.

Roger Joseph Boscovich in 1755 based in his work on the shape of the earth proposed in his bookDe Litteraria expeditione per pontificiam ditionem ad dimetiendos duos meridiani gradus a PP. Maire et Boscovicli that the true value of a series of observations would be that which minimises the sum of absolute errors. In modern terminology this value is the median. The first example of what later became known as the normal curve was studied byAbraham de Moivre who plotted this curve on November 12, 1733.^[17] de Moivre was studying the number of heads that occurred when a 'fair' coin was tossed.

In 1763 Richard Price transmitted to the Royal SocietyThomas Bayes proof of a rule for using a binomial distribution to calculate a posterior probability on a prior event.

In 1765Joseph Priestley invented the firsttimeline charts.

Johann Heinrich Lambert in his 1765 bookAnlage zur Architectonic proposed thesemicircle as a distribution of errors:

${\displaystyle f(x)=\frac{1}{2}\sqrt{(1-x^2)}}$

with -1 <x < 1.

Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve and deduced a formula for the mean of three observations.

Laplace in 1774 noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded.^[18][19] This distribution is now known as theLaplace distribution. Lagrange proposed aparabolic fractal distribution of errors in 1776.

Laplace in 1778 published his second law of errors wherein he noted that the frequency of an error was proportional to the exponential of the square of its magnitude. This was subsequently rediscovered byGauss (possibly in 1795) and is now best known as thenormal distribution which is of central importance in statistics.^[20] This distribution was first referred to as thenormal distribution byC. S. Peirce in 1873 who was studying measurement errors when an object was dropped onto a wooden base.^[21] He chose the termnormal because of its frequent occurrence in naturally occurring variables.

Lagrange also suggested in 1781 two other distributions for errors – araised cosine distribution and alogarithmic distribution.

Laplace gave (1781) a formula for the law of facility of error (a term due toJoseph Louis Lagrange, 1774), but one which led to unmanageable equations.Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

In 1786William Playfair (1759–1823) introduced the idea of graphical representation into statistics. He invented theline chart,bar chart andhistogram and incorporated them into his works oneconomics, theCommercial and Political Atlas. This was followed in 1795 by his invention of thepie chart and circle chart which he used to display the evolution of England's imports and exports. These latter charts came to general attention when he published examples in hisStatistical Breviary in 1801.

Laplace, in an investigation of the motions ofSaturn andJupiter in 1787, generalized Mayer's method by using different linear combinations of a single group of equations.

In 1791Sir John Sinclair introduced the term 'statistics' into English in hisStatistical Accounts of Scotland.

In 1802 Laplace estimated the population of France to be 28,328,612.^[22] He calculated this figure using the number of births in the previous year and census data for three communities. The census data of these communities showed that they had 2,037,615 persons and that the number of births were 71,866. Assuming that these samples were representative of France, Laplace produced his estimate for the entire population.

Themethod of least squares, which was used to minimize errors in datameasurement, was published independently byAdrien-Marie Legendre (1805),Robert Adrain (1808), andCarl Friedrich Gauss (1809). Gauss had used the method in his famous 1801 prediction of the location of thedwarf planetCeres. The observations that Gauss based his calculations on were made by the Italian monk Piazzi.

The method of least squares was preceded by the use a median regression slope. This method minimizing the sum of the absolute deviances. A method of estimating this slope was invented byRoger Joseph Boscovich in 1760 which he applied to astronomy.

The termprobable error (der wahrscheinliche Fehler) – the median deviation from the mean – was introduced in 1815 by the German astronomerFrederik Wilhelm Bessel.Antoine Augustin Cournot in 1843 was the first to use the termmedian (valeur médiane) for the value that divides a probability distribution into two equal halves.

Other contributors to the theory of errors were Ellis (1844),De Morgan (1864),Glaisher (1872), andGiovanni Schiaparelli (1875).^{[citation needed]} Peters's (1856) formula for${\displaystyle r}$, the "probable error" of a single observation was widely used and inspired earlyrobust statistics (resistant tooutliers: seePeirce's criterion).

In the 19th century authors onstatistical theory included Laplace,S. Lacroix (1816), Littrow (1833),Dedekind (1860), Helmert (1872),Laurent (1873), Liagre, Didion,De Morgan andBoole.

Gustav Theodor Fechner used the median (Centralwerth) in sociological and psychological phenomena.^[23] It had earlier been used only in astronomy and related fields.Francis Galton used the English termmedian for the first time in 1881 having earlier used the termsmiddle-most value in 1869 and themedium in 1880.^[24]

Adolphe Quetelet (1796–1874), another important founder of statistics, introduced the notion of the "average man" (l'homme moyen) as a means of understanding complex social phenomena such ascrime rates,marriage rates, andsuicide rates.^[25]

The first tests of the normal distribution were invented by the German statisticianWilhelm Lexis in the 1870s. The only data sets available to him that he was able to show were normally distributed were birth rates.

Although the origins of statistical theory lie in the 18th-century advances in probability, the modern field of statistics only emerged in the late-19th and early-20th century in three stages. The first wave, at the turn of the century, was led by the work ofFrancis Galton andKarl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. The second wave of the 1910s and 20s was initiated byWilliam Sealy Gosset, and reached its culmination in the insights ofRonald Fisher. This involved the development of betterdesign of experiments models, hypothesis testing and techniques for use with small data samples. The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work betweenEgon Pearson andJerzy Neyman in the 1930s.^[26] Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology.

The first statistical bodies were established in the early 19th century. TheRoyal Statistical Society was founded in 1834 andFlorence Nightingale, its first female member, pioneered the application of statistical analysis to health problems for the furtherance of epidemiological understanding and public health practice. However, the methods then used would not be considered as modern statistics today.

TheOxford scholarFrancis Ysidro Edgeworth's book,Metretike: or The Method of Measuring Probability and Utility (1887) dealt with probability as the basis of inductive reasoning, and his later works focused on the 'philosophy of chance'.^[27] His first paper on statistics (1883) explored the law of error (normal distribution), and hisMethods of Statistics (1885) introduced an early version of thet distribution, theEdgeworth expansion, theEdgeworth series, the method of variate transformation and the asymptotic theory of maximum likelihood estimates.

The NorwegianAnders Nicolai Kiær introduced the concept ofstratified sampling in 1895.^[28]Arthur Lyon Bowley introduced new methods of data sampling in 1906 when working on social statistics. Although statistical surveys of social conditions had started withCharles Booth's "Life and Labour of the People in London" (1889–1903) andSeebohm Rowntree's "Poverty, A Study of Town Life" (1901), Bowley's key innovation consisted of the use ofrandom sampling techniques. His efforts culminated in hisNew Survey of London Life and Labour.^[29]

Francis Galton is credited as one of the principal founders of statistical theory. His contributions to the field included introducing the concepts ofstandard deviation,correlation,regression and the application of these methods to the study of the variety of human characteristics – height, weight, eyelash length among others. He found that many of these could be fitted to a normal curve distribution.^[30]

Galton submitted a paper toNature in 1907 on the usefulness of the median.^[31] He examined the accuracy of 787 guesses of the weight of an ox at a country fair. The actual weight was 1208 pounds: the median guess was 1198. The guesses were markedly non-normally distributed (cf.Wisdom of the Crowd).

Galton's publication ofNatural Inheritance in 1889 sparked the interest of a brilliant mathematician,Karl Pearson,^[32] then working atUniversity College London, and he went on to found the discipline of mathematical statistics.^[33] He emphasised the statistical foundation of scientific laws and promoted its study and his laboratory attracted students from around the world attracted by his new methods of analysis, includingUdny Yule. His work grew to encompass the fields ofbiology,epidemiology, anthropometry,medicine and socialhistory. In 1901, withWalter Weldon, founder ofbiometry, and Galton, he founded the journalBiometrika as the first journal of mathematical statistics and biometry.

His work, and that of Galton, underpins many of the 'classical' statistical methods which are in common use today, including theCorrelation coefficient, defined as a product-moment;^[34] themethod of moments for the fitting of distributions to samples;Pearson's system of continuous curves that forms the basis of the now conventional continuous probability distributions;Chi distance a precursor and special case of theMahalanobis distance^[35] andP-value, defined as the probability measure of the complement of theball with the hypothesized value as center point and chi distance as radius.^[35] He also introduced the term 'standard deviation'.

He also founded thestatistical hypothesis testing theory,^[35]Pearson's chi-squared test andprincipal component analysis.^[36][37] In 1911 he founded the world's first university statistics department atUniversity College London.

The second wave of mathematical statistics was pioneered byRonald Fisher who wrote two textbooks,Statistical Methods for Research Workers, published in 1925 andThe Design of Experiments in 1935, that were to define the academic discipline in universities around the world. He also systematized previous results, putting them on a firm mathematical footing. In his 1918 seminal paperThe Correlation between Relatives on the Supposition of Mendelian Inheritance, the first use to use the statistical term,variance. In 1919, atRothamsted Experimental Station he started a major study of the extensive collections of data recorded over many years. This resulted in a series of reports under the general titleStudies in Crop Variation. In 1930 he publishedThe Genetical Theory of Natural Selection where he applied statistics toevolution.

Over the next seven years, he pioneered the principles of thedesign of experiments (see below) and elaborated his studies of analysis of variance. He furthered his studies of the statistics of small samples. Perhaps even more important, he began his systematic approach of the analysis of real data as the springboard for the development of new statistical methods. He developed computational algorithms for analyzing data from his balanced experimental designs. In 1925, this work resulted in the publication of his first book,Statistical Methods for Research Workers.^[38] This book went through many editions and translations in later years, and it became the standard reference work for scientists in many disciplines. In 1935, this book was followed byThe Design of Experiments, which was also widely used.

In addition to analysis of variance, Fisher named and promoted the method ofmaximum likelihood estimation. Fisher also originated the concepts ofsufficiency,ancillary statistics,Fisher's linear discriminator andFisher information. His articleOn a distribution yielding the error functions of several well known statistics (1924) presentedPearson's chi-squared test andWilliam Sealy Gosset'st in the same framework as theGaussian distribution, and his own parameter in the analysis of varianceFisher's z-distribution (more commonly used decades later in the form of theF distribution).^[39]The 5% level of significance appears to have been introduced by Fisher in 1925.^[40] Fisher stated that deviations exceeding twice the standard deviation are regarded as significant. Before this deviations exceeding three times the probable error were considered significant. For a symmetrical distribution the probable error is half the interquartile range. For a normal distribution the probable error is approximately 2/3 the standard deviation. It appears that Fisher's 5% criterion was rooted in previous practice.

Other important contributions at this time includedCharles Spearman'srank correlation coefficient that was a useful extension of the Pearson correlation coefficient.William Sealy Gosset, the English statistician better known under his pseudonym ofStudent, introducedStudent's t-distribution, a continuous probability distribution useful in situations where the sample size is small and population standard deviation is unknown.

Egon Pearson (Karl's son) andJerzy Neyman introduced the concepts of "Type II" error, power of a test andconfidence intervals.Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling.^[41]

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In 1747, while serving as surgeon on HM BarkSalisbury,James Lind carried out a controlled experiment to develop a cure forscurvy.^[42] In this study his subjects' cases "were as similar as I could have them", that is he provided strict entry requirements to reduce extraneous variation. The men were paired, which providedblocking. From a modern perspective, the main thing that is missing is randomized allocation of subjects to treatments.

Lind is today often described as a one-factor-at-a-time experimenter.^[43] Similar one-factor-at-a-time (OFAT) experimentation was performed at theRothamsted Research Station in the 1840s by SirJohn Lawes to determine the optimal inorganic fertilizer for use on wheat.^[43]

A theory of statistical inference was developed byCharles S. Peirce in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883), two publications that emphasized the importance of randomization-based inference in statistics. In another study, Peirce randomly assigned volunteers to ablinded,repeated-measures design to evaluate their ability to discriminate weights.^{[44][45][46][47]}

Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s.^{[44][45][46][47]} Peirce also contributed the first English-language publication on anoptimal design forregression-models in 1876.^[48] A pioneeringoptimal design forpolynomial regression was suggested byGergonne in 1815.^{[citation needed]} In 1918Kirstine Smith published optimal designs for polynomials of degree six (and less).^[49]

The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, was pioneered^[50] byAbraham Wald in the context of sequential tests of statistical hypotheses.^[51] Surveys are available of optimalsequential designs,^[52] and ofadaptive designs.^[53] One specific type of sequential design is the "two-armed bandit", generalized to themulti-armed bandit, on which early work was done byHerbert Robbins in 1952.^[54]

The term "design of experiments" (DOE) derives from early statistical work performed bySir Ronald Fisher. He was described byAnders Hald as "a genius who almost single-handedly created the foundations for modern statistical science."^[55] Fisher initiated the principles ofdesign of experiments and elaborated on his studies of "analysis of variance". Perhaps even more important, Fisher began his systematic approach to the analysis of real data as the springboard for the development of new statistical methods. He began to pay particular attention to the labour involved in the necessary computations performed by hand, and developed methods that were as practical as they were founded in rigour. In 1925, this work culminated in the publication of his first book,Statistical Methods for Research Workers.^[56] This went into many editions and translations in later years, and became a standard reference work for scientists in many disciplines.^[57]

A methodology for designing experiments was proposed byRonald A. Fisher, in his innovative bookThe Design of Experiments (1935) which also became a standard.^{[58][59][60][61]} As an example, he described how to test thehypothesis that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. While this sounds like a frivolous application, it allowed him to illustrate the most important ideas of experimental design: seeLady tasting tea.

Agricultural science advances served to meet the combination of larger city populations and fewer farms. But for crop scientists to take due account of widely differing geographical growing climates and needs, it was important to differentiate local growing conditions. To extrapolate experiments on local crops to a national scale, they had to extend crop sample testing economically to overall populations. As statistical methods advanced (primarily the efficacy of designed experiments instead of one-factor-at-a-time experimentation), representative factorial design of experiments began to enable the meaningful extension, by inference, of experimental sampling results to the population as a whole.^{[citation needed]} But it was hard to decide how representative was the crop sample chosen.^{[citation needed]} Factorial design methodology showed how to estimate and correct for any random variation within the sample and also in the data collection procedures.

The termBayesian refers toThomas Bayes (1702–1761), who proved that probabilistic limits could be placed on an unknown event. However it wasPierre-Simon Laplace (1749–1827) who introduced (as principle VI) what is now calledBayes' theorem and applied it tocelestial mechanics, medical statistics,reliability, andjurisprudence.^[62] When insufficient knowledge was available to specify an informed prior, Laplace useduniform priors, according to his "principle of insufficient reason".^[62][63] Laplace assumed uniform priors for mathematical simplicity rather than for philosophical reasons.^[62] Laplace also introduced^{[citation needed]} primitive versions ofconjugate priors and thetheorem ofvon Mises andBernstein, according to which the posteriors corresponding to initially differing priors ultimately agree, as the number of observations increases.^[64] This early Bayesian inference, which used uniform priors following Laplace'sprinciple of insufficient reason, was called "inverse probability" (because itinfers backwards from observations to parameters, or from effects to causes^[65]).

After the 1920s,inverse probability was largely supplanted^{[citation needed]} by a collection of methods that were developed byRonald A. Fisher,Jerzy Neyman andEgon Pearson. Their methods came to be calledfrequentist statistics.^[65] Fisher rejected the Bayesian view, writing that "the theory of inverse probability is founded upon an error, and must be wholly rejected".^[66] At the end of his life, however, Fisher expressed greater respect for the essay of Bayes, which Fisher believed to have anticipated his own,fiducial approach to probability; Fisher still maintained that Laplace's views on probability were "fallacious rubbish".^[66] Neyman started out as a "quasi-Bayesian", but subsequently developedconfidence intervals (a key method in frequentist statistics) because "the whole theory would look nicer if it were built from the start without reference to Bayesianism and priors".^[67]The wordBayesian appeared around 1950, and by the 1960s it became the term preferred by those dissatisfied with the limitations of frequentist statistics.^[65][68]

In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise toobjective andsubjective currents in Bayesian practice. In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.^[69] No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.

In the further development of Laplace's ideas, subjective ideas predate objectivist positions. The idea that 'probability' should be interpreted as 'subjective degree of belief in a proposition' was proposed, for example, byJohn Maynard Keynes in the early 1920s.^{[citation needed]} This idea was taken further byBruno de Finetti in Italy (Fondamenti Logici del Ragionamento Probabilistico, 1930) andFrank Ramsey in Cambridge (The Foundations of Mathematics, 1931).^[70] The approach was devised to solve problems with the frequentist definition of probability but also with the earlier, objectivist approach of Laplace.^[69] The subjective Bayesian methods were further developed and popularized in the 1950s byL.J. Savage.^{[citation needed]}

Objective Bayesian inference was further developed byHarold Jeffreys at theUniversity of Cambridge. His bookTheory of Probability first appeared in 1939 and played an important role in the revival of theBayesian view of probability.^[71][72] In 1957,Edwin Jaynes promoted the concept ofmaximum entropy for constructing priors, which is an important principle in the formulation of objective methods, mainly for discrete problems. In 1965,Dennis Lindley's two-volume work "Introduction to Probability and Statistics from a Bayesian Viewpoint" brought Bayesian methods to a wide audience. In 1979,José-Miguel Bernardo introducedreference analysis,^[69] which offers a general applicable framework for objective analysis.^[73] Other well-known proponents of Bayesian probability theory includeI.J. Good,B.O. Koopman,Howard Raiffa,Robert Schlaifer andAlan Turing.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery ofMarkov chain Monte Carlo methods, which removed many of thecomputational problems, and an increasing interest in nonstandard, complex applications.^[74] Despite growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.^[75] Nonetheless, Bayesian methods are widely accepted and used, such as for example in the field ofmachine learning.^[76]

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• Pearson, Egon (1978).The History of Statistics in the 17th and 18th Centuries against the changing background of intellectual, scientific and religious thought (Lectures by Karl Pearson given at University College London during the academic sessions 1921-1933). New York: MacMillan Publishing Co., Inc. p. 744.ISBN 978-0-02-850120-8.
• Salsburg, David (2001).The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century.ISBN 0-7167-4106-7
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• JEHPS: Recent publications in the history of probability and statistics
• Electronic Journ@l for History of Probability and Statistics/Journ@l Electronique d'Histoire des Probabilités et de la Statistique
• Figures from the History of Probability and Statistics (Univ. of Southampton)
• Materials for the History of Statistics (Univ. of York)
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
• Earliest Uses of Symbols in Probability and Statistics onEarliest Uses of Various Mathematical Symbols

• History of probability and statistics
• History of science by discipline

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