Frequentist probability and choice under uncertainty

Advances in mathematics education, 2014

This chapter presents a twenty first century historical and philosophical perspective on probability, related to the teaching of probability. It is important to remember the historical development as it provides pointers to be taken into account in developing a modern curriculum in teaching probability at all levels. We include some elements relating to continuous as well as discrete distributions. Starting with initial ideas of chance two millennia ago, we move on to the correspondence of Pascal and Fermat, and insurance against risk. Two centuries of debate and discussion led to the key fundamental ideas; the twentieth century saw the climax of the axiomatic approach from Kolmogorov. Philosophical difficulties have been prevalent in probability since its inception, especially since the idea requires modelling-probability is not an inherent property of an event, but is based on the underlying model chosen. Hence the arguments about the philosophical basis of probability have still not been fully resolved. The three main theories (APT, FQT, and SJT) are described, relating to the symmetric, frequentist, and subjectivist approaches. These philosophical ideas are key to developing teaching content and methodology. Probabilistic concepts are closer to a consistent way of thinking about the world rather than describing the world in a consistent manner, which seems paradoxical, and can only be resolved by a careful analysis.

The purpose of the paper is to introduce the evolution of modern probability from its birth in the seventeenth century onwards.

Philosophies

Probability as understood today, namely as a quantitative notion expressible by means of a function ranging in the interval between 0-1, took shape in the mid-17th century, and presents both a mathematical and a philosophical aspect. Of these two sides, the second is by far the most controversial, and fuels a heated debate, still ongoing. After a short historical sketch of the birth and developments of probability, its major interpretations are outlined, by referring to the work of their most prominent representatives. The final section addresses the question of whether any of such interpretations can presently be considered predominant, which is answered in the negative.

Advances in Pure Mathematics, 2012

Classical statistics and Bayesian statistics refer to the frequentist and subjective theories of probability respectively. Von Mises and De Finetti, who authored those conceptualizations, provide interpretations of the probability that appear incompatible. This discrepancy raises ample debates and the foundations of the probability calculus emerge as a tricky, open issue so far. Instead of developing philosophical discussion, this research resorts to analytical and mathematical methods. We present two theorems that sustain the validity of both the frequentist and the subjective views on the probability. Secondly we show how the double facets of the probability turn out to be consistent within the present logical frame.

Archive for History of Exact Sciences, 1994

We discuss the evolution of an idea which contains, within the setting of an urn model, the notion of a martingale. The idea is to be found in PO~SSON (1837) but its main proponent is CATALAN in a series of papers beginning in 1841, in partial ignorance of PoIssoN's work. The usual BAYESIAN coloration is present. A letter from BIENAYMt~ of 1878, possibly his last, to CATALAN elucidates the origin of the idea, and illustrates the personal relations of French probabilists at the time.

We show that the dominant definitions of probability are seriously flawed. While finite frequentism fails to define, infinite frequentism is not operational. Similarly, the Dutch Book arguments used to establish the existence of subjective probability fail to do so. An alternative definition of probability as a metaphor is offered. It is shown that this definition resolves several puzzles regarding the interpretation of common frequentist procedures and also some puzzles regarding the philosophy of science.

Revista Sergipana de Matemática e Educação Matemática, 2021

We focus on the ways in which we can use a frequentist interpretation of probability to develop suitable methods for statistical inference. The discussion about the controversy in the foundations reveals that a frequentist conception is highly prone to dispute, as a justification of this view fails from a rational perspective when the explication of probability integrates statistical inference. We give an overview on the dispute and the crucial examples that highlight the deficiencies of a purely frequentist position towards probability. The concept of probability emerges from a mixture of classical, frequentist, and subjectivist meanings, which are not easy to separate. A shift in connotation of probability towards a biased frequentist meaning decreases the scope of probability or the quality of applications. Probability is a complementary concept, which falls apart if we reduce it to one view. This gives rise to investigate refined approaches towards teaching from a wider perspective on the range of meanings of probability apart from frequentist aspects. Empirical studies show the shortcomings of educational approaches that ignore subjectivist aspects of probability, which leads to far-reaching misconceptions not only about the use of Bayes' formula but also in the perception of probabilities at large. Resumo Nesse texto temos como foco a discussão sobre as maneiras pelas quais uma interpretação frequentista da probabilidade pode ser usada para desenvolver métodos adequados para inferência estatística. O debate sobre a controvérsia sobre os fundamentos revela que uma concepção frequentistaé altamente propensa a argumentos como uma justificativa desse ponto de vista falha em uma perspectiva racional quando a explicação da probabilidade integra a inferência estatística. Damos uma visão geral do debate e dos exemplos cruciais que destacam as fragilidades de uma posição puramente frequentista em relaçãoà probabilidade. O conceito de probabilidade emerge de uma mistura de significados clássicos, frequentistas e subjetivistas, que não são fáceis de separar. A mudança na conotação da probabilidade para um significado frequentista tendencioso

BSHM Bulletin: Journal of the British Society for the History of Mathematics, 2018

The paper discusses the background to and provides a transcription of a letter from Robert Leslie Ellis (1817-59) to William Walton (1813Walton ( -1901) ) of 1849 on probability theory. A lthough today still a largely forgotten figure, the polymath Robert Leslie Ellis (1817-59) was an important member of the Cambridge mathematical community in the mid-nineteenth century. 1 As a pupil of William Hopkins, student of George Peacock, First Wrangler in 1840, editor of the Cambridge Mathematical Journal and frequent contributor to the important British mathematical journals of the time, Ellis was much admired by his Victorian contemporaries. George Boole, in an 1857 prize-essay about applications of probability theory, wrote that '[t]here' is no living mathematician for whose intellectual character I entertain a more sincere respect than I do for that of Mr. Ellis' (Boole 1857 [1952], 350). A decade later, Francis Galton expressed a similar sentiment when describing Ellis as one 'whose name is familiar to generations of Cambridge men as a prodigy of universal genius' (Galton 1869 [1892], 18). During his tragically short life, Ellis wrote five papers and one note on probability theory (1844a [1863], 1844b [1863], 1844c, 1850a, 1850b, 1854 [1863]). 2 As the Reverend Harvey Goodwin noted in his biographical memoir accompanying The Mathematical and Other Writings of Robert Leslie Ellis, Ellis' papers on probability represented 'as well as possible his special taste [...] with regard to mathematics' (Goodwin 1863, xxix). Given his 'speculative mind', Ellis, who always talked about probability 'with great pleasure' and as a subject in which he was 'thoroughly at home', was naturally drawn to questions of foundations (Goodwin 1863, xxix, xxxv). Ellis' contributions to mathematical probability theory, written in the 1840s to 1850s, roughly consisted of two parts: firstly, an early sketch of a frequentist theory of probability and, secondly, a simplification and extension to any number of unknowns of Pierre-Simon Laplace's demonstration of the method of least squares. 3

During the Synod of Constance (Konstanz) (1414 -1418) the controversial topic of probabilism was publicly discussed probably for the first time in history. But only one hundred years later the Spanish Dominican Bartholomé de Medina (1527 -1581) led the way to probabilism as an acknowledged principle in the moral teaching of the Catholic Church. Soon afterwards probabilism was taken up by the Jesuits and subsequently further developed to a formal theory. In 1662 the book La Logique ou l'Art de Penser (in Latin Ars cogitandi) was published anonymously in Paris. In this influential book the concept of probability is used in the context of chance. Since then, numerous scientists have given various interpretations of the notion of probability, which invariably proved to be ineligible. In this paper the historical development of the notion probability is sketched and, finally, based on Jakob Bernoulli's ideas, a sustainable solution is presented.