Bayes' theorem
In probability theory and statistics, Bayesâ theorem (alternatively Bayesâ law or Bayes' rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayesâ theorem, a personâs age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age.
One of the many applications of Bayesâ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayesâ theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics.
Bayesâ theorem is named after Rev. Thomas Bayes (/ËbeÉȘz/; 1701â1761), who first provided an equation that allows new evidence to update beliefs. It was further developed by Pierre-Simon Laplace, who first published the modern formulation in his 1812 âThĂ©orie analytique des probabilitĂ©s.â Sir Harold Jeffreys put Bayesâ algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayesâ theorem âis to the theory of probability what the Pythagorean theorem is to geometry.â[1]
Statement of theorem
Bayes' theorem is stated mathematically as the following equation:[2]
where and are events and .
- and are the probabilities of observing and without regard to each other.
- , a conditional probability, is the probability of observing event given that is true.
- is the probability of observing event given that is true.
History
Bayesâ theorem was named after the Reverend Thomas Bayes (1701â1761), who studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). Bayesâ unpublished manuscript was significantly edited by Richard Price before it was posthumously read at the Royal Society. Price edited[3] Bayesâ major work âAn Essay towards solving a Problem in the Doctrine of Chancesâ (1763), which appeared in âPhilosophical Transactions,â[4] and contains Bayesâ Theorem. Price wrote an introduction to the paper which provides some of the philosophical basis of Bayesian statistics. In 1765 he was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes.[5][6]
The French mathematician Pierre-Simon Laplace reproduced and extended Bayesâ results in 1774, apparently quite unaware of Bayesâ work.[7][8] The Bayesian interpretation of probability was developed mainly by Laplace.[9]
Stephen Stigler suggested in 1983 that Bayesâ theorem was discovered by Nicholas Saunderson, a blind English mathematician, some time before Bayes;[10][11] that interpretation, however, has been disputed.[12] Martyn Hooper[13] and Sharon McGrayne[14] have argued that Richard Price's contribution was substantial:
By modern standards, we should refer to the BayesâPrice rule. Price discovered Bayesâ work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayesâ name alone is unfair but so entrenched that anything else makes little sense.[14]
Examples
Drug testing
Suppose a drug test is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a positive test is a user?
Despite the apparent accuracy of the test, if an individual tests positive, it is more likely that they do not use the drug than that they do. This surprising result arises because the number of non-users is very large compared to the number of users; thus the number of false positives outweighs the number of true positives. To use concrete numbers, if 1000 individuals are tested, there are expected to be 995 non-users and 5 users. From the 995 non-users, 0.01 Ă 995 â 10 false positives are expected. From the 5 users, 0.99 Ă 5 â 5 true positives are expected. Out of 15 positive results, only 5, about 33%, are genuine. This illustrates the importance of base rates, and how the formation of policy can be egregiously misguided if base rates are neglected.[15]
The importance of specificity in this example can be seen by calculating that even if sensitivity is raised to 100% and specificity remains at 99% then the probability of the person being a drug user only rises from 33.2% to 33.4%, but if the sensitivity is held at 99% and the specificity is increased to 99.5% then probability of the person being a drug user rises to about 49.9%.
A more complicated example
The entire output of a factory is produced on three machines. The three machines account for different amounts of the factory output, namely 20%, 30%, and 50%. The fraction of defective items produced is this: for the first machine, 5%; for the second machine, 3%; for the third machine, 1%. If an item is chosen at random from the total output and is found to be defective, what is the probability that it was produced by the third machine?
A solution is as follows. Let Ai denote the event that a randomly chosen item was made by the ith machine (for i = 1,2,3). Let B denote the event that a randomly chosen item is defective. Then, we are given the following information:
- P(A1) = 0.2, P(A2) = 0.3, P(A3) = 0.5.
If the item was made by the first machine, then the probability that it is defective is 0.05; that is, P(B | A1) = 0.05. Overall, we have
- P(B | A1) = 0.05, P(B | A2) = 0.03, P(B | A3) = 0.01.
To answer the original question, we first find P(B). That can be done in the following way:
- P(B) = ÎŁi P(B | Ai)âŻP(Ai) = (0.05)(0.2) + (0.03)(0.3) + (0.01)(0.5) = 0.024.
Hence 2.4% of the total output of the factory is defective.
We are given that B has occurred, and we want to calculate the conditional probability of A3. By Bayes' theorem,
- P(A3 | B) = P(B | A3)âŻP(A3)/P(B) = (0.01)(0.50)/(0.024) = 5/24.
Given that the item is defective, the probability that it was made by the third machine is only 5/24. Although machine 3 produces half of the total output, it produces a much smaller fraction of the defective items. Hence the knowledge that the item selected was defective enables us to replace the prior probability P(A3) = 1/2 by the smaller posterior probability P(A3 | B) = 5/24.
Once again, the answer can be reached without recourse to the formula by applying the conditions to any hypothetical number of cases. For example, if 100,000 items are produced by the factory, 20,000 will be produced by Machine A, 30,000 by Machine B, and 50,000 by Machine C. Machine A will produce 1000 defective items, Machine B 900, and Machine C 500. Of the total 2400 defective items, only 500, or 5/24 were produced by Machine C.
Interpretations
The interpretation of Bayesâ theorem depends on the interpretation of probability ascribed to the terms. The two main interpretations are described below.
Bayesian interpretation
In the Bayesian (or epistemological) interpretation, probability measures a âdegree of belief.â Bayesâ theorem then links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If the coin is flipped a number of times and the outcomes observed, that degree of belief may rise, fall or remain the same depending on the results.
For proposition A and evidence B,
- Pâ(Aâ), the prior, is the initial degree of belief in A.
- Pâ(A | Bâ), the âposterior,â is the degree of belief having accounted for B.
- the quotient P(B |A )/P(B) represents the support B provides for A.
For more on the application of Bayesâ theorem under the Bayesian interpretation of probability, see Bayesian inference.
Frequentist interpretation
In the frequentist interpretation, probability measures a âproportion of outcomes.â For example, suppose an experiment is performed many times. P(A) is the proportion of outcomes with property A, and P(B) that with property B. P(B | Aâ) is the proportion of outcomes with property B out of outcomes with property A, and P(A | Bâ) the proportion of those with A out of those with B.
The role of Bayesâ theorem is best visualized with tree diagrams, as shown to the right. The two diagrams partition the same outcomes by A and B in opposite orders, to obtain the inverse probabilities. Bayesâ theorem serves as the link between these different partitionings.
Example
An entomologist spots what might be a rare subspecies of beetle, due to the pattern on its back. In the rare subspecies, 98% have the pattern, or P(Pattern | Rare) = 98%. In the common subspecies, 5% have the pattern. The rare subspecies accounts for only 0.1% of the population. How likely is the beetle having the pattern to be rare, or what is P(Rare | Pattern)?
From the extended form of Bayesâ theorem (since any beetle can be only rare or common),
Forms
Events
Simple form
For events A and B, provided that P(B) â 0,
In many applications, for instance in Bayesian inference, the event B is fixed in the discussion, and we wish to consider the impact of its having been observed on our belief in various possible events A. In such a situation the denominator of the last expression, the probability of the given evidence B, is fixed; what we want to vary is A. Bayesâ theorem then shows that the posterior probabilities are proportional to the numerator:
- (proportionality over A for given B).
In words: posterior is proportional to prior times likelihood.[16]
If events A1, A2, ..., are mutually exclusive and exhaustive, i.e., one of them is certain to occur but no two can occur together, and we know their probabilities up to proportionality, then we can determine the proportionality constant by using the fact that their probabilities must add up to one. For instance, for a given event A, the event A itself and its complement ÂŹA are exclusive and exhaustive. Denoting the constant of proportionality by c we have
Adding these two formulas we deduce that
or
Alternative form
Another form of Bayesâ Theorem that is generally encountered when looking at two competing statements or hypotheses is:
For an epistemological interpretation:
For proposition A and evidence or background B,[17]
- P(A),the prior probability, is the initial degree of belief in A.
- P(âA), is the corresponding probability of the initial degree of belief against A: 1 â P(A) = P(âA)
- P(B | A), the conditional probability or likelihood, is the degree of belief in B, given that the proposition A is true.
- P(B | âA), the conditional probability or likelihood, is the degree of belief in B, given that the proposition A is false.
- P(A | B), the posterior probability, is the probability for A after taking into account B for and against A.
Extended form
Often, for some partition {Aj} of the sample space, the event space is given or conceptualized in terms of P(Aj) and P(B | Aj). It is then useful to compute P(B) using the law of total probability:
In the special case where A is a binary variable:
Random variables
Consider a sample space Ω generated by two random variables X and Y. In principle, Bayesâ theorem applies to the events A = {X = x} and B = {Y = y}. However, terms become 0 at points where either variable has finite probability density. To remain useful, Bayesâ theorem may be formulated in terms of the relevant densities (see Derivation).
Simple form
If X is continuous and Y is discrete,
If X is discrete and Y is continuous,
If both X and Y are continuous,
Extended form
A continuous event space is often conceptualized in terms of the numerator terms. It is then useful to eliminate the denominator using the law of total probability. For fY(y), this becomes an integral:
Bayesâ rule
Bayesâ theorem in odds form is:
where
is called the Bayes factor or likelihood ratio and the odds between two events is simply the ratio of the probabilities of the two events. Thus
So the rule says that the posterior odds are the prior odds times the Bayes factor, or in other words, posterior is proportional to prior times likelihood.
Derivation
For events
Bayesâ theorem may be derived from the definition of conditional probability:
because
For random variables
For two continuous random variables X and Y, Bayesâ theorem may be analogously derived from the definition of conditional density:
See also
Notes
- â Jeffreys, Harold (1973). Scientific Inference (3rd ed.). Cambridge University Press. p. 31. ISBN 978-0-521-18078-8.
- â Stuart, A.; Ord, K. (1994), Kendall's Advanced Theory of Statistics: Volume IâDistribution Theory, Edward Arnold, §8.7.
- â Richard Allen (1999). David Hartley on Human Nature. SUNY Press. pp. 243â4. ISBN 978-0-7914-9451-6. Retrieved 16 June 2013.
- â Bayes, Thomas & Price, Richard (1763). "An Essay towards solving a Problem in the Doctrine of Chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S." (PDF). Philosophical Transactions of the Royal Society of London. 53 (0): 370â418. doi:10.1098/rstl.1763.0053.
- â Holland, pp. 46â7.
- â Richard Price (1991). Price: Political Writings. Cambridge University Press. p. xxiii. ISBN 978-0-521-40969-8. Retrieved 16 June 2013.
- â Laplace refined Bayesâ theorem over a period of decades:
- Laplace announced his independent discovery of Bayes' theorem in: Laplace (1774) âMĂ©moire sur la probabilitĂ© des causes par les Ă©vĂ©nements,â âMĂ©moires de l'AcadĂ©mie royale des Sciences de MI (Savants Ă©trangers),â 4: 621â656. Reprinted in: Laplace, âOeuvres complĂštesâ (Paris, France: Gauthier-Villars et fils, 1841), vol. 8, pp. 27â65. Available on-line at: Gallica. Bayesâ theorem appears on p. 29.
- Laplace presented a refinement of Bayesâ theorem in: Laplace (read: 1783 / published: 1785) âMĂ©moire sur les approximations des formules qui sont fonctions de trĂšs grands nombres,â âMĂ©moires de l'AcadĂ©mie royale des Sciences de Paris,â 423â467. Reprinted in: Laplace, âOeuvres complĂštesâ (Paris, France: Gauthier-Villars et fils, 1844), vol. 10, pp. 295â338. Available on-line at: Gallica. Bayesâ theorem is stated on page 301.
- See also: Laplace, âEssai philosophique sur les probabilitĂ©sâ (Paris, France: Mme. Ve. Courcier [Madame veuve (i.e., widow) Courcier], 1814), page 10. English translation: Pierre Simon, Marquis de Laplace with F. W. Truscott and F. L. Emory, trans., âA Philosophical Essay on Probabilitiesâ (New York, New York: John Wiley & Sons, 1902), page 15.
- â Daston, Lorraine (1988). Classical Probability in the Enlightenment. Princeton Univ Press. p. 268. ISBN 0-691-08497-1.
- â Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Harvard University Press, Chapter 3.
- â Stigler, Stephen M (1983). "Who Discovered Bayesâ Theorem?". The American Statistician. 37 (4): 290â296. doi:10.1080/00031305.1983.10483122.
- â De Vaux, Richard; Velleman, Paul; Bock, David (2016). Stats, Data and Models (4 ed.). Pearson. pp. 380â381. ISBN 978-0-321-98649-8.
- â Edwards, A. W. F. (1986). "Is the Reference in Hartley (1749) to Bayesian Inference?". The American Statistician. 40 (2): 109â110. doi:10.1080/00031305.1986.10475370.
- â Hooper, Martyn (2013). "Richard Price, Bayesâ theorem, and God". Significance. 10 (1): 36â39. doi:10.1111/j.1740-9713.2013.00638.x.
- 1 2 McGrayne, S. B. (2011). The Theory That Would Not Die: How Bayesâ Rule Cracked the Enigma Code, Hunted Down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy. Yale University Press. ISBN 978-0-300-18822-6.
- â Daniel Kahneman (25 October 2011). Thinking, Fast and Slow. Macmillan. ISBN 978-1-4299-6935-2. Retrieved 8 April 2012.
- â Lee, Peter M. (2012). "Chapter 1". Bayesian Statistics. Wiley. ISBN 978-1-1183-3257-3.
- â "Bayesâ Theorem: Introduction". Trinity University.
Further reading
- Bruss, F. Thomas (2013), â250 years of âAn Essay towards solving a Problem in the Doctrine of Chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S.â,â doi:10.1365/s13291-013-0077-z, Jahresbericht der Deutschen Mathematiker-Vereinigung, Springer Verlag, Vol. 115, Issue 3-4 (2013), 129-133.
- Gelman, A, Carlin, JB, Stern, HS, and Rubin, DB (2003), âBayesian Data Analysis,â Second Edition, CRC Press.
- Grinstead, CM and Snell, JL (1997), âIntroduction to Probability (2nd edition),â American Mathematical Society (free pdf available) .
- Hazewinkel, Michiel, ed. (2001) [1994], "Bayes formula", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- McGrayne, SB (2011). The Theory That Would Not Die: How Bayesâ Rule Cracked the Enigma Code, Hunted Down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy. Yale University Press. ISBN 978-0-300-18822-6.
- Laplace, P (1774/1986), âMemoir on the Probability of the Causes of Eventsâ, Statistical Science 1(3):364â378.
- Lee, Peter M (2012), âBayesian Statistics: An Introduction,â 4th edition. Wiley. ISBN 978-1-118-33257-3.
- Puga JL, Krzywinski M, Altman N (31 March 2015). "Bayesâ theorem". Nature Methods. 12 (4): 277â8.
- Rosenthal, Jeffrey S (2005), âStruck by Lightning: The Curious World of Probabilities.â HarperCollins. (Granta, 2008. ISBN 9781862079960).
- Stigler, SM (1986). "Laplaceâs 1774 Memoir on Inverse Probability". Statistical Science. 1 (3): 359â363. doi:10.1214/ss/1177013620.
- Stone, JV (2013), download chapter 1 of âBayesâ Rule: A Tutorial Introduction to Bayesian Analysisâ, Sebtel Press, England.
- Bayesian Reasoning for Intelligent People, An introduction and tutorial to the use of Bayes' theorem in statistics and cognitive science.
External links
- Bayesâ theorem at EncyclopĂŠdia Britannica
- The Theory That Would Not Die by Sharon Bertsch McGrayne New York Times Book Review by John Allen Paulos on 5 August 2011
- Visual explanation of Bayes using trees (video)
- Bayesâ frequentist interpretation explained visually (video)
- Earliest Known Uses of Some of the Words of Mathematics (B). Contains origins of âBayesian,â âBayesâ Theorem,â âBayes Estimate/Risk/Solution,â âEmpirical Bayes,â and âBayes Factor.â
- Weisstein, Eric W. "Bayesâ Theorem". MathWorld.
- Bayesâ theorem at PlanetMath.org.
- Bayes Theorem and the Folly of Prediction
- A tutorial on probability and Bayesâ theorem devised for Oxford University psychology students
- An Intuitive Explanation of Bayes' Theorem by Eliezer S. Yudkowsky