In probability theory and statistics, Bayes’s theorem (alternatively Bayes’s law or Bayes’s 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 Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer than can be done without knowledge of the person’s age.
One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics.
Bayes’ theorem is named after Reverend Thomas Bayes (//; 1701?–1761), who first used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter, published as An Essay towards solving a Problem in the Doctrine of Chances (1763). In what he called a scholium, Bayes extended his algorithm to any unknown prior cause. Independently of Bayes, Pierre-Simon Laplace in 1774, and later in his 1812 Théorie analytique des probabilités, used conditional probability to formulate the relation of an updated posterior probability from a prior probability, given evidence. Sir Harold Jeffreys put Bayes's algorithm and Laplace’s formulation on an axiomatic basis. Jeffreys wrote that Bayes’ theorem “is to the theory of probability what the Pythagorean theorem is to geometry.”
Statement of theorem
where and are events and .
Suppose that a test for using a particular drug 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 drug user?
Even if an individual tests positive, it is more likely that they do not use the drug than that they do. This is because the number of non-users is large compared to the number of users. The number of false positives outweighs the number of true positives. For example, if 1000 individuals are tested, there are expected to be 995 non-users and 5 users. From the 995 non-users, 0.01 × 995 ≃ 10 false positives are expected. From the 5 users, 0.99 × 5 ≈ 5 true positives are expected. Out of 15 positive results, only 5 are genuine.
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 the probability of the person being a drug user rises to about 49.9%.
Even if 100% of patients with pancreatic cancer have a certain symptom, when someone has the same symptom, it does not mean that this person has a 100% chance of getting pancreatic cancer. Assume the incidence rate is 1/100000, and 1/10000 non-cancer people with the same symptoms worldwide, the probability of having pancreatic cancer is only 9%, and the rest 91% may be false positive
The table presents the corresponding numbers per 100,000 people.
A more complicated example
The entire output of a factory is produced on three machines. The three machines account for 20%, 30%, and 50% of the factory output. The fraction of defective items produced is 5% for the first machine; 3% for the second machine; and 1% for the third machine. 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?
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.
A solution is as follows. Let Xi denote the event that a randomly chosen item was made by the i th machine (for i = A,B,C). Let Y denote the event that a randomly chosen item is defective. Then, we are given the following information:
If the item was made by the first machine, then the probability that it is defective is 0.05; that is, P(Y | XA) = 0.05. Overall, we have
To answer the original question, we first find P(Y). That can be done in the following way:
Hence 2.4% of the total output of the factory is defective.
We are given that Y has occurred, and we want to calculate the conditional probability of XC. By Bayes’ theorem,
Given that the item is defective, the probability that it was made by the third machine is only 5/24. Although machine C 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(XC) = 1/2 by the smaller posterior probability P(XC | Y) = 5/24.
The interpretation of Bayes’ theorem depends on the interpretation of probability ascribed to the terms. The two main interpretations are described below.
In the Bayesian (or epistemological) interpretation, probability measures a “degree of belief.” Bayes’ 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,
- P (A), the prior, is the initial degree of belief in A.
- P (A | B), the posterior is the degree of belief having accounted for B.
- the quotient P(B | A)/ represents the support B provides for A.
For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see Bayesian inference.
In the frequentist interpretation, probability measures a “proportion of outcomes.” 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 | A) is the proportion of outcomes with property B out of outcomes with property A, and P(A | B) the proportion of those with A out of those with B.
The role of Bayes’ 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. Bayes' theorem serves as the link between these different partitionings.
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 Bayes’ theorem (since any beetle can be only rare or common),
For events A and B, provided that P(B) ≠ 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. Bayes’ theorem then shows that the posterior probabilities are proportional to the numerator:
- (proportionality over A for given B).
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 ¬A are exclusive and exhaustive. Denoting the constant of proportionality by c we have
Adding these two formulas we deduce that
Another form of Bayes’ theorem that is generally encountered when looking at two competing statements or hypotheses is:
For an epistemological interpretation:
- is the prior probability, is the initial degree of belief in A.
- is the corresponding probability of the initial degree of belief against A, where
- is the conditional probability or likelihood, is the degree of belief in B, given that the proposition A is true.
- is the conditional probability or likelihood, is the degree of belief in B, given that the proposition A is false.
- is the posterior probability, is the probability for A after taking into account B for and against A.
In the special case where A is a binary variable:
If X is continuous and Y is discrete,
where each is a density function.
If X is discrete and Y is continuous,
If both X and Y are continuous,
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:
Bayes’ theorem in odds form is:
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.
In the special case that and , one writes , and uses a similar abbreviation for the Bayes factor and for the conditional odds. The odds on is by definition the odds for and against . Bayes’ rule can then be written in the abbreviated form
or in words: the posterior odds on equals the prior odds on times the likelihood ratio for given information . In short, posterior odds equals prior odds times likelihood ratio.
Bayes’ theorem may be derived from the definition of conditional probability:
where is the joint probability of both A and B being true, because
Correspondence to other mathematical frameworks
The corresponding formula in terms of probability calculus is Bayes' theorem which in its expanded form is expressed as:
In the equation above the conditional probability generalizes the logical statement , i.e. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. The term denotes the prior probability (aka. the base rate) of . Assume that is equivalent to being TRUE, and that is equivalent to being FALSE. It is then easy to see that when i.e. when is TRUE. This is because so that the fraction on the right-hand side of the equation above is equal to 1, and hence which is equivalent to being TRUE. Hence, Bayes’ theorem represents a generalization of contraposition.
Bayes’ theorem represents a special case of conditional inversion in subjective logic expressed as:
where denotes the operator for conditional inversion. The argument denotes a pair of binomial conditional opinions given by source , and the argument denotes the prior probability (aka. the base rate) of . The pair of inverted conditional opinions is denoted . The conditional opinion generalizes the probabilistic conditional , i.e. in addition to assigning a probability the source can assign any subjective opinion to the conditional statement . A binomial subjective opinion is the belief in the truth of statement with degrees of uncertainty, as expressed by source . Every subjective opinion has a corresponding projected probability . The projected probability of opinions applied to Bayes’ theorem produces a homomorphism so that Bayes’ theorem can be expressed in terms of the projected probabilities of opinions:
Using the chain rule
And, on the other hand
The desired result is obtained by identifying both expressions and solving for .
Bayes’ theorem was named after Thomas Bayes (1701–1761), who studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). Bayes’s unpublished manuscript was significantly edited by Richard Price before it was posthumously read at the Royal Society. Price edited Bayes’s major work “An Essay towards solving a Problem in the Doctrine of Chances” (1763), which appeared in Philosophical Transactions, and contains Bayes’ 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.
The French mathematician Pierre-Simon Laplace reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work. The Bayesian interpretation of probability was developed mainly by Laplace.
Stephen Stigler used a Bayesian argument to conclude that Bayes’ theorem was discovered by Nicholas Saunderson, a blind English mathematician, some time before Bayes; that interpretation, however, has been disputed. Martyn Hooper and Sharon McGrayne have argued that Richard Price's contribution was substantial:
By modern standards, we should refer to the Bayes–Price rule. Price discovered Bayes’ work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes’ name alone is unfair but so entrenched that anything else makes little sense.
- Laplace refined Bayes' theorem over a period of decades:
- Laplace announced his independent discovery of Bayes' theorem in: Laplace (1774) "Mémoire sur la probabilité des causes par les événements," "Mémoires de l'Académie royale des Sciences de MI (Savants étrangers)," 4: 621–656. Reprinted in: Laplace, "Oeuvres complètes" (Paris, France: Gauthier-Villars et fils, 1841), vol. 8, pp. 27–65. Available on-line at: Gallica. Bayes' theorem appears on p. 29.
- Laplace presented a refinement of Bayes' theorem in: Laplace (read: 1783 / published: 1785) "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres," "Mémoires de l'Académie royale des Sciences de Paris," 423–467. Reprinted in: Laplace, "Oeuvres complètes" (Paris, France: Gauthier-Villars et fils, 1844), vol. 10, pp. 295–338. Available on-line at: Gallica. Bayes' theorem is stated on page 301.
- See also: Laplace, "Essai philosophique sur les probabilités" (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., "A Philosophical Essay on Probabilities" (New York, New York: John Wiley & Sons, 1902), page 15.
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