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{{DISPLAYTITLE:''An Essay towards solving a Problem in the Doctrine of Chances''}}
'''''An Essay towards solving a Problem in the Doctrine of Chances''''' is a work on the mathematical [[theory of probability]] by the Reverend [[Thomas Bayes]], published in 1763,<ref name=Bayes1/> two years after its author's death.  It included a statement of a special case of what is now called [[Bayes' theorem]].  In 18th-century English, the phrase "doctrine of chances" meant the theory of probability.  It had been introduced as [[The Doctrine of Chances|the title of a book]] by [[Abraham de Moivre]].


Bayes supposed a sequence of independent experiments, each having as its outcome either success or failure, the probability of success being some number ''p'' between 0 and&nbsp;1.  But then he supposed ''p'' to be an uncertain quantity, whose probability of being in any interval between 0 and 1 is the length of the interval.  In modern terms, ''p'' would be considered a [[random variable]] [[uniform distribution (continuous)|uniformly distributed]] between 0 and&nbsp;1.  [[Conditional distribution|Conditionally on the value of ''p'']], the trials resulting in success or failure are independent, but unconditionally (or "[[marginal distribution|marginally]]") they are not.  That is because if a large number of successes are observed, then ''p'' is more likely to be large, so that success on the next trial is more probable.  The question Bayes addressed was: what is the conditional probability distribution of ''p'', given the numbers of successes and failures so far observed.  The answer is that its [[probability density function]] is


: <math> f(p) = \frac{(n+1)!}{k!(n-k)!} p^k (1-p)^{n-k}\text{ for }0\le p \le 1  </math>
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(and ''&fnof;''(''p'')&nbsp;=&nbsp;0 for ''p''&nbsp;<&nbsp;0 or ''p''&nbsp;>&nbsp;1) where ''k'' is the number of successes so far observed, and ''n'' is the number of trials so far observed.  This is what today is called the [[Beta distribution]] with parameters ''k''&nbsp;+&nbsp;1 and ''n''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;1.
 
==Outline==
 
Bayes' preliminary results (Propositions 3, 4, and 5) imply the truth of the theorem that is named for him. Particularly, Proposition 5 gives a simple description of [[conditional probability]]:
 
:''"If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event has also happened, from hence I guess that the first event has also happened, the probability I am right is P/b."''
 
However, it does not appear that Bayes emphasized or focused on this finding. He presented his work as the solution to a problem:
 
:''"<nowiki>Given the number of times in which an unknown event has happened and failed [... Find] the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.</nowiki>"''<ref name="Price1763"/>
 
Bayes gave an example of a man trying to guess the ratio of "blanks" and "prizes" at a lottery. So far the man has watched the lottery draw ten blanks and one prize. Given these data, Bayes showed in detail how to compute the probability that the ratio of blanks to prizes is between 9:1 and 11:1 (the probability is low - about 7.7%). He went on to describe that computation after the man has watched the lottery draw twenty blanks and two prizes, forty blanks and four prizes, and so on. Finally, having drawn 10,000 blanks and 1,000 prizes, the probability reaches about 97%.<ref name="Price1763"/>
 
Bayes' main result (Proposition 9) is the following in modern terms:
 
:Assume a [[prior distribution#Uninformative priors|uniform prior distribution]] of the binomial parameter <math>p</math>. After observing <math>m</math> successes and <math>n</math> failures,
:: <math>
P(a<p<b|m;n)=
\frac {\int_a^b {n+m \choose m} p^m (1-p)^n\,dp}
{\int_0^1 {n+m \choose m} p^m (1-p)^n\,dp}.
\!</math>
 
It is unclear whether Bayes was a "Bayesian" in the modern sense. That is, whether he was interested in [[inference|Bayesian inference]], or merely in [[probability]]. Proposition 9 seems "Bayesian" in its presentation as a probability about the [[parameter]] <math>p</math>. However, Bayes stated his question in a manner that suggests a frequentist viewpoint: he supposed that a ball is thrown at random onto a square table (this table is often misrepresented as a billiard table, and the ball as a billiard ball, but Bayes never describes them as such), and considered further balls that fall to the left or right of the first ball with probabilities <math>p</math> and <math>1-p</math>. The algebra is of course identical no matter which view is taken.
 
===Richard Price and the existence of God===
[[Richard Price]] discovered Bayes' essay and its now-famous theorem in Bayes' papers after Bayes' death. He believed that Bayes' Theorem helped prove the existence of [[Deity|God ("the Deity")]] and wrote the following in his introduction to the essay:
 
:''"The purpose I mean is, to shew what reason we have for believing that there are in the constitution of things fixt laws according to which things happen, and that, therefore, the frame of the world must be the effect of the wisdom and power of an intelligent cause; and thus to confirm the argument taken from final causes for the existence of the Deity. It will be easy to see that the converse problem solved in this essay is more directly applicable to this purpose; for it shews us, with distinctness and precision, in every case of any particular order or recurrency of events, what reason there is to think that such recurrency or order is derived from stable causes or regulations in nature, and not from any irregularities of chance."'' (Philosophical Transactions of the Royal Society of London, 1763)<ref name="Price1763"/>
 
In modern terms this is an instance of the [[teleological argument]].
 
===Versions of the essay===
* {{Cite journal
|last=Bayes |first=Thomas
|year=1763
|title = An Essay towards solving a Problem in the Doctrine of Chances.
|journal = [[Philosophical Transactions of the Royal Society of London]]
|volume = 53
|issue=0
|pages = 370–418
|doi = 10.1098/rstl.1763.0053
|last2=Price
|first2=Mr.
}}
* {{cite journal
|first=G |last=Barnard
|year=1958
|title=Studies in the History of Probability and Statistics: IX. Thomas Bayes's ''[[An Essay Towards Solving a Problem in the Doctrine of Chances]]''
|journal=[[Biometrika]]
|volume=45 |issue=3–4 |pages=296–315. |doi=10.1093/biomet/45.3-4.293
}}
* Thomas Bayes [http://www.stat.ucla.edu/history/essay.pdf "An Essay towards solving a Problem in the Doctrine of Chances"]. ''(Bayes' essay in the original notation)''
 
===Commentaries===
* [[George Alfred Barnard|G. A. Barnard]] (1958) "Studies in the History of Probability and Statistics: IX. Thomas Bayes' Essay Towards Solving a Problem in the Doctrine of Chances", ''Biometrika'' 45:293–295. ''(biographical remarks)''
* Stephen M. Stigler (1982). "Thomas Bayes' Bayesian Inference," ''Journal of the Royal Statistical Society'', Series A, 145:250–258. ''(Stigler argues for a revised interpretation of the essay; recommended)''
* [[Isaac Todhunter]] (1865). ''A History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace'', Macmillan. Reprinted 1949, 1956 by Chelsea and 2001 by Thoemmes.
 
==References==
{{reflist|refs=
 
<ref name=Bayes1>[[Thomas Bayes|Bayes T.]] (1763) [http://rstl.royalsocietypublishing.org/content/53/370.full.pdf "An Essay towards solving a Problem in the Doctrine of Chances"]. ''Phil. Trans.'', 53, 370–418. {{doi|10.1098/rstl.1763.0053}}</ref>
 
<ref name="Price1763">
{{cite journal |doi = 10.1098/rstl.1763.0053 |journal = Philosophical Transactions of the Royal Society of London | volume = 53 |issue = 0 | year = 1763 | pages = 370–418 | url = http://www.stat.ucla.edu/history/essay.pdf | title = An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, communicated  by Mr. Price, in a letter to John Canton, M. A. and F. R. S. | author = Bayes, Thomas, and Price, Richard }}</ref>  
}}
 
== External links ==
* [http://www.stat.ucla.edu/history/essay.pdf ''An Essay towards solving a Problem in the Doctrine of Chances''] at the [[UCLA]] Department of Statistics
 
{{DEFAULTSORT:Essay towards solving a Problem in the Doctrine of Chances, An}}
[[Category:History of statistics]]
[[Category:History of mathematics]]
[[Category:Bayesian inference]]
[[Category:1763 works]]
[[Category:18th-century essays]]

Latest revision as of 20:28, 20 December 2014


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Would you like to eat a lot much less red meat? Then, try using it as being a condiment. It can be used to aid include flavoring and feel to organic or grain-centered dishes. Chinese and Mediterranean ethnicities try this and also have lower costs of heart-related diseases.

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Consume prepared food items as opposed to fried ones. Generally food items that are baked will be more healthy and have much less carbohydrate food, unhealthy natural oils and calories. Also, they won't zap your power throughout the day the way in which deep fried foods are apt to do.

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