|
|
| Line 1: |
Line 1: |
| {{Redirect|Statistical probability| the episode of Star Trek: Deep Space Nine|Statistical Probabilities}}
| | 56 yrs old Carpenter and Joiner Issac from Fairview, has hobbies and interests such as football, free coins fifa 14 hack and textiles. Last month just made vacation to Citadel of the Ho Dynasty.<br><br>Feel free to visit my homepage; [https://youtube.com/watch?v=HSoNAvTIeEc fifa 14 hack ipad] |
| {{multiple issues|
| |
| {{Refimprove|date=March 2008}}
| |
| {{technical|date=February 2013}}
| |
| }}
| |
| | |
| [[Image:John Venn.jpg|thumb|150px|John Venn]]
| |
| | |
| '''Frequentist probability''' or '''frequentism''' is the standard [[interpretation of probability]]; it defines an event's [[probability]] as the [[limit of a sequence|limit]] of its relative [[frequency]] in a large number of trials.
| |
| | |
| The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the [[Classical definition of probability|classical interpretation]]. In the classical interpretation, probability was defined in terms of the [[principle of indifference]], based on the natural symmetry of a problem, so, ''e.g.'' the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning.
| |
| | |
| The shift from the classical view to the frequentist view represents a [[paradigm shift]] in the progression of statistical thought.{{citation needed|date=April 2012}} This school is often associated with the names of [[Jerzy Neyman]] and [[Egon Pearson]] who described the logic of [[statistical hypothesis testing]].{{citation needed|date=April 2012}} Other influential figures of the frequentist school{{clarify|reason=what school is this|date=April 2012}} include [[John Venn]], [[Ronald Aylmer Fisher|R.A. Fisher]], and [[Richard von Mises]].<ref name="Gilles">''The Frequency theory'' Chapter 5; discussed in Donald Gilles, ''Philosophical theories of probability'' (2000), Psychology Press. ISBN 9780415182751 , p. 88.</ref>
| |
| | |
| ==Definition==
| |
| In the frequentist interpretation, probabilities are discussed only when dealing with well-defined [[experiment (probability theory)|random experiments]].{{Citation needed|date=July 2008}} The [[Set (mathematics)|set]] of all possible outcomes of a random experiment is called the [[sample space]] of the experiment. An [[event (probability theory)|event]] is defined as a particular [[subset]] of the sample space to be considered. For any given event, only one of two possibilities may hold: it occurs or it does not. The [[relative frequency]] of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the '''probability''' of that event. This is the core conception of probability in the frequentist interpretation.
| |
| | |
| Thus, if <math>n_t</math> is the total number of trials and <math>n_x</math> is the number of trials where the event <math>x</math> occurred, the probability <math>P(x)</math> of the event occurring will be approximated by the relative frequency as follows:
| |
| | |
| :<math>P(x) \approx \frac{n_x}{n_t}.</math>
| |
| | |
| Clearly, as the number of trials is increased, one might expect the relative frequency to become a better approximation of a "true frequency".
| |
| | |
| A controversial{{citation needed|date=December 2012}} claim of the frequentist approach is that in the "long run," as the number of trials approaches infinity, the relative frequency will converge ''exactly'' to the true probability:<ref>von Mises, Richard (1939) ''Probability, Statistics, and Truth'' (in German) (English translation, 1981: Dover Publications; 2 Revised edition. ISBN 0486242145) (p.14)</ref>
| |
| | |
| :<math>P(x) = \lim_{n_t\rightarrow \infty}\frac{n_x}{n_t}.</math>
| |
| | |
| Such a limit is possible only in theory (''e.g.'' counting the relative fraction of even numbers less than ''n''<sub>t</sub>: one may easily compute the limit <math>n_t\to\infty</math>.) This conflicts with the standard claim{{citation needed|date=April 2012}} that the frequency interpretation is somehow more "objective" than other theories of probability.
| |
| | |
| == Scope ==
| |
| | |
| The frequentist interpretation is a philosophical approach to the definition and use of probabilities; it is one of several, and, historically, the earliest to challenge the classical interpretation.{{citation needed|date=April 2012}} It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages.
| |
| | |
| As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations. It offers distinct guidance in the construction and design of practical experiments, especially when contrasted with the [[Bayesian probability|Bayesian interpretation]]. As to whether this guidance is useful, or is apt to mis-interpretation, has been a source of controversy. Particularly when the frequency interpretation of probability is mistakenly assumed to be the only possible basis for [[frequentist inference]]. So, for example, a list of mis-interpretations of the meaning of [[p-values]] accompanies the article on p-values; controversies are detailed in the article on [[Statistical hypothesis testing#Controversy|statistical hypothesis testing]]. The [[Jeffreys–Lindley paradox]] shows how different interpretations, applied to the same data set, can lead to different conclusions about the 'statistical significance' of a result.{{citation needed|date=April 2012}}
| |
| | |
| As [[William Feller]] noted:<ref>[[William Feller]] (1957), ''An Introduction to Probability Theory and Its Applications, Vol. 1'', page 4</ref>
| |
| {{Quote|There is no place in our system for speculations concerning the probability that the [[sunrise problem|sun will rise tomorrow]]. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines "out of infinitely many worlds one is selected at random..." Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.}}
| |
| | |
| == History ==
| |
| {{main|History of probability}}
| |
| {{Expand section|date=December 2008}}
| |
| The frequentist view was arguably{{citation needed|date=April 2012}} foreshadowed by [[Aristotle]], in ''[[Rhetoric (Aristotle)|Rhetoric]]'',<ref name="keynesVIII">[[John Maynard Keynes|Keynes, John Maynard]]; ''A Treatise on Probability'' (1921), Chapter VIII “The Frequency Theory of Probability”.</ref> when he wrote:
| |
| | |
| {{Quote|the probable is that which for the most part happens<ref name="aristorhetor">''Rhetoric'' Bk 1 Ch 2; discussed in J. Franklin, ''The Science of Conjecture: Evidence and Probability Before Pascal'' (2001), The Johns Hopkins University Press. ISBN 0801865697 , p. 110.</ref>}}
| |
| | |
| It was given explicit statement by [[Robert Leslie Ellis]] in "On the Foundations of the Theory of Probabilities"<ref name="ellisfound">Ellis, Robert Leslie (1843) “On the Foundations of the Theory of Probabilities”, ''Transactions of the Cambridge Philosophical Society'' vol 8</ref> read on 14 February 1842,<ref name="keynesVIII" /> (and much later again in "Remarks on the Fundamental Principles of the Theory of Probabilities"<ref name="ellisfund">Ellis, Robert Leslie (1854) “Remarks on the Fundamental Principles of the Theory of Probabilitiess”, ''Transactions of the Cambridge Philosophical Society'' vol 9</ref>). [[Antoine Augustin Cournot]] presented the same conception in 1843, in ''Exposition de la théorie des chances et des probabilités''.<ref>Cournot, Antoine Augustin (1843) ''Exposition de la théorie des chances et des probabilités''. L. Hachette, Paris.
| |
| [http://www.archive.org/details/expositiondelat00courgoog archive.org]</ref>
| |
| | |
| Perhaps the first elaborate and systematic exposition{{citation needed|date=April 2012}} was by [[John Venn]],<ref name=Venn/> in ''The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability'' (published editions in 1866, 1876, 1888).
| |
| | |
| ==Etymology==
| |
| According to the [[Oxford English Dictionary]], the term 'frequentist' was first used by [[Maurice Kendall|M. G. Kendall]] in 1949, to contrast with [[Bayesian probability|Bayesians]], whom he called "non-frequentists".<ref>[http://www.leidenuniv.nl/fsw/verduin/stathist/1stword.htm Earliest Known Uses of Some of the Words of Probability & Statistics]</ref><ref>{{cite journal
| |
| |last=Kendall
| |
| |first=Maurice George
| |
| |author-link=Maurice Kendall
| |
| |title=On the Reconciliation of Theories of Probability
| |
| |journal=Biometrika
| |
| |year=1949
| |
| |volume=36
| |
| |pages=101–116
| |
| |issue=1/2
| |
| |publisher=Biometrika Trust
| |
| |jstor=2332534 |doi=10.1093/biomet/36.1-2.101
| |
| }}</ref> He observed
| |
| :3....we may broadly distinguish two main attitudes. One takes probability as 'a degree of rational belief', or some similar idea...the second defines probability in terms of frequencies of occurrence of events, or by relative proportions in 'populations' or 'collectives'; (p. 101)
| |
| :...
| |
| :12. It might be thought that the differences between the frequentists and the non-frequentists (if I may call them such) are largely due to the differences of the domains which they purport to cover. (p. 104)
| |
| :...
| |
| :''I assert that this is not so'' ... The essential distinction between the frequentists and the non-frequentists is, I think, that the former, in an effort to avoid anything savouring of matters of opinion, seek to define probability in terms of the objective properties of a population, real or hypothetical, whereas the latter do not. [emphasis in original]
| |
| | |
| ==Alternative views==
| |
| {{Main|Probability interpretations}}
| |
| {{Expand section|date=April 2012}}
| |
| The frequentist interpretation does resolve difficulties with the classical interpretation, such as any problem where the natural symmetry of outcomes is not known. It does not address other issues, such as the [[dutch book]]. [[Propensity probability]] is an alternative physicalist approach.{{citation needed|date=April 2012}}
| |
| | |
| ==Notes==
| |
| {{reflist|refs=
| |
| | |
| <ref name=Venn>Venn, John (1888) ''The Logic of Chance'', 3rd Edition [http://www.archive.org/details/logicofchance029416mbp archive.org]. Full title: ''The Logic of Chance: An essay on the foundations and province of the theory of probability, with especial reference to its logical bearings and its application to Moral and Social Science, and to Statistics'', Macmillan & Co, London
| |
| </ref>
| |
| | |
| }}
| |
| | |
| ==References==
| |
| * P W Bridgman, ''The Logic of Modern Physics'', 1927
| |
| * Alonzo Church, ''The Concept of a Random Sequence'', 1940
| |
| * Harald Cramér, ''Mathematical Methods of Statistics'', 1946
| |
| * William Feller, ''An introduction to Probability Theory and its Applications'', 1957
| |
| * P Martin-Löf, ''On the Concept of a Random Sequence'', 1966
| |
| * Richard von Mises, ''Probability, Statistics, and Truth'', 1939 (German original 1928)
| |
| * Jerzy Neyman, ''First Course in Probability and Statistics'', 1950
| |
| * Hans Reichenbach, ''The Theory of Probability'', 1949 (German original 1935)
| |
| * Bertrand Russell, ''Human Knowledge'', 1948
| |
| * {{cite doi|10.1006/aama.1999.0653}} [http://www.ma.utexas.edu/~friedman/freq.ps PS]
| |
| | |
| {{statistics}}
| |
| | |
| {{Use dmy dates|date=January 2011}}
| |
| | |
| {{DEFAULTSORT:Frequency Probability}}
| |
| [[Category:Probability interpretations]]
| |
56 yrs old Carpenter and Joiner Issac from Fairview, has hobbies and interests such as football, free coins fifa 14 hack and textiles. Last month just made vacation to Citadel of the Ho Dynasty.
Feel free to visit my homepage; fifa 14 hack ipad