HOME

TheInfoList



OR:

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...
, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole. One of the many applications of Bayes' theorem is
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
, a particular approach to
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
. When applied, the probabilities involved in the theorem may have different probability interpretations. With
Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...
interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesian statistics, being considered "to the theory of probability what
Pythagoras's theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
is to geometry."


Statement of theorem

Bayes' theorem is stated mathematically as the following equation: where A and B are events and P(B) \neq 0. * P(A\mid B) is a conditional probability: the probability of event A occurring given that B is true. It is also called the
posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
of A given B. * P(B\mid A) is also a conditional probability: the probability of event B occurring given that A is true. It can also be interpreted as the likelihood of A given a fixed B because P(B\mid A)=L(A\mid B). * P(A) and P(B) are the probabilities of observing A and B respectively without any given conditions; they are known as the prior probability and
marginal probability In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variab ...
.


Proof


For events

Bayes' theorem may be derived from the definition of conditional probability: :P(A\mid B)=\frac, \text P(B) \neq 0, where P(A \cap B) is the probability of both A and B being true. Similarly, :P(B\mid A)=\frac, \text P(A) \neq 0, Solving for P(A \cap B) and substituting into the above expression for P(A\mid B) yields Bayes' theorem: :P(A\mid B) = \frac, \text P(B) \neq 0.


For continuous random variables

For two continuous random variables ''X'' and ''Y'', Bayes' theorem may be analogously derived from the definition of
conditional density In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the c ...
: :f_ (x) = \frac :f_(y) = \frac Therefore, :f_(x) = \frac.


General case

Let P_Y^x be the conditional distribution of Y given X = x and let P_X be the distribution of X. The joint distribution is then P_ (dx,dy) = P_Y^x (dy) P_X (dx). The conditional distribution P_X^y of X given Y=y is then determined by P_X^y (A) = E (1_A (X) , Y = y) Existence and uniqueness of the needed
conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
is a consequence of the Radon-Nikodym theorem. This was formulated by Kolmogorov in his famous book from 1933. Kolmogorov underlines the importance of conditional probability by writing 'I wish to call attention to ... and especially the theory of conditional probabilities and conditional expectations ...' in the Preface. The Bayes theorem determines the posterior distribution from the prior distribution. Bayes theorem can be generalized to include improper prior distributions such as the uniform distribution on the real line. Modern Markov Chain Monte Carlo methods have boosted the importance of Bayes theorem including cases with improper priors.


Examples


Recreational mathematics

Bayes' rule and computing conditional probabilities provide a solution method for a number of popular puzzles, such as the Three Prisoners problem, the
Monty Hall problem The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show '' Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solve ...
, the Two Child problem and the Two Envelopes problem.


Drug testing

Suppose, a particular test for whether someone has been using cannabis is 90% sensitive, meaning the true positive rate (TPR) = 0.90. Therefore, it leads to 90% true positive results (correct identification of drug use) for cannabis users. The test is also 80% specific, meaning true negative rate (TNR) = 0.80. Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or false positive rate (FPR) = 0.20, for non-users. Assuming 0.05 prevalence, meaning 5% of people use cannabis, what is the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
that a random person who tests positive is really a cannabis user? The
Positive predictive value The positive and negative predictive values (PPV and NPV respectively) are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV des ...
(PPV) of a test is the proportion of persons who are actually positive out of all those testing positive, and can be calculated from a sample as: :PPV = True positive / Tested positive If sensitivity, specificity, and prevalence are known, PPV can be calculated using Bayes theorem. Let P(\text\mid \text) mean "the probability that someone is a cannabis user given that they test positive," which is what is meant by PPV. We can write: : \begin P(\text\mid \text) &= \frac \\ &= \frac \\ pt &= \frac = \frac \approx 19\% \end The fact that P(\text) = P(\text\mid\text) P(\text) + P(\text\mid\text) P(\text) is a direct application of the Law of Total Probability. In this case, it says that the probability that someone tests positive is the probability that a user tests positive, times the probability of being a user, plus the probability that a non-user tests positive, times the probability of being a non-user. This is true because the classifications user and non-user form a partition of a set, namely the set of people who take the drug test. This combined with the definition of conditional probability results in the above statement. In other words, even if someone tests positive, the probability that they are a cannabis user is only 19% — this is because in this group, only 5% of people are users, and most positives are false positives coming from the remaining 95%. If 1,000 people were tested: * 950 are non-users and 190 of them give false positive (0.20 × 950) * 50 of them are users and 45 of them give true positive (0.90 × 50) The 1,000 people thus yields 235 positive tests, of which only 45 are genuine drug users, about 19%. See Figure 1 for an illustration using a frequency box, and note how small the pink area of true positives is compared to the blue area of false positives.


Sensitivity or specificity

The importance of specificity can be seen by showing that even if sensitivity is raised to 100% and specificity remains at 80%, the probability of someone testing positive really being a cannabis user only rises from 19% to 21%, but if the sensitivity is held at 90% and the specificity is increased to 95%, the probability rises to 49%.


Cancer rate

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. Assuming the incidence rate of pancreatic cancer is 1/100000, while 10/99999 healthy individuals have the same symptoms worldwide, the probability of having pancreatic cancer given the symptoms is only 9.1%, and the other 90.9% could be "false positives" (that is, falsely said to have cancer; "positive" is a confusing term when, as here, the test gives bad news). Based on incidence rate, the following table presents the corresponding numbers per 100,000 people. Which can then be used to calculate the probability of having cancer when you have the symptoms: : \begin P(\text, \text) &= \frac \\ &= \frac \\ pt&= \frac = \frac1 \approx 9.1\% \end


Defective item rate

A factory produces items using three machines—A, B, and C—which account for 20%, 30%, and 50% of its output respectively. Of the items produced by machine A, 5% are defective; similarly, 3% of machine B's items and 1% of machine C's are defective. If a randomly selected item is defective, what is the probability it was produced by machine C? Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases. For example, if the factory produces 1,000 items, 200 will be produced by Machine A, 300 by Machine B, and 500 by Machine C. Machine A will produce 5% × 200 = 10 defective items, Machine B 3% × 300 = 9, and Machine C 1% × 500 = 5, for a total of 24. Thus, the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%). This problem can also be solved using Bayes' theorem: 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: :P(X_A) = 0.2, \quad P(X_B) = 0.3, \quad P(X_C) = 0.5. If the item was made by the first machine, then the probability that it is defective is 0.05; that is, ''P''(''Y'' ,  ''X''A) = 0.05. Overall, we have :P(Y, X_A) = 0.05, \quad P(Y , X_B) = 0.03, \quad P(Y, X_C) = 0.01. To answer the original question, we first find ''P''(Y). That can be done in the following way: :P(Y) = \sum_i P(Y, X_i) P(X_i) = (0.05)(0.2) + (0.03)(0.3) + (0.01)(0.5) = 0.024. Hence, 2.4% of the total output is defective. We are given that ''Y'' has occurred, and we want to calculate the conditional probability of ''X''C. By Bayes' theorem, :P(X_C, Y) = \frac = \frac = \frac Given that the item is defective, the probability that it was made by machine C is 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''(''X''C) = 1/2 by the smaller posterior probability ''P''(XC ,  ''Y'') = 5/24.


Interpretations

The interpretation of Bayes' rule depends on the interpretation of probability ascribed to the terms. The two main interpretations are described below. Figure 2 shows a geometric visualization.


Bayesian interpretation

In the Bayesian (or epistemological) interpretation, probability measures a "degree of belief". Bayes' theorem 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 will probably rise or fall, but might even 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 after incorporating news that ''B'' is true. * the quotient represents the support ''B'' provides for ''A''. For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
.


Frequentist interpretation

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'' (the prior) and ''P''(''B'') is the proportion with property ''B''. ''P''(''B'' ,  ''A'') is the proportion of outcomes with property ''B'' ''out of'' outcomes with property ''A'', and ''P''(''A'' ,  ''B'') is the proportion of those with ''A'' ''out of'' those with ''B'' (the posterior). The role of Bayes' theorem is best visualized with tree diagrams such as Figure 3. The two diagrams partition the same outcomes by ''A'' and ''B'' in opposite orders, to obtain the inverse probabilities. Bayes' theorem links the different partitionings.


Example

An
entomologist Entomology () is the scientific study of insects, a branch of zoology. In the past the term "insect" was less specific, and historically the definition of entomology would also include the study of animals in other arthropod groups, such as ara ...
spots what might, due to the pattern on its back, be a rare
subspecies In biological classification, subspecies is a rank below species, used for populations that live in different areas and vary in size, shape, or other physical characteristics ( morphology), but that can successfully interbreed. Not all specie ...
of
beetle Beetles are insects that form the order Coleoptera (), in the superorder Endopterygota. Their front pair of wings are hardened into wing-cases, elytra, distinguishing them from most other insects. The Coleoptera, with about 400,000 describ ...
. A full 98% of the members of the rare subspecies have the pattern, so ''P''(Pattern ,  Rare) = 98%. Only 5% of members of the common subspecies have the pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is ''P''(Rare ,  Pattern)? From the extended form of Bayes' theorem (since any beetle is either rare or common), : \begin P(\text \mid \text) &= \frac \\ pt&= \frac \\ pt&= \frac \\ pt&\approx 1.9\% \end


Forms


Events


Simple form

For events ''A'' and ''B'', provided that ''P''(''B'') ≠ 0, :P(A, B) = \frac . In many applications, for instance in
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
, 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, so the last equation becomes: :P(A, B) \propto P(A) \cdot P(B, A) . In words, the posterior is proportional to the prior times the likelihood. If events ''A''1, ''A''2, ..., are mutually exclusive and exhaustive, i.e., one of them is certain to occur but no two can occur together, 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 :P(A, B) = c \cdot P(A) \cdot P(B, A) \text P(\neg A, B) = c \cdot P(\neg A) \cdot P(B, \neg A). Adding these two formulas we deduce that : 1 = c \cdot (P(B, A)\cdot P(A) + P(B, \neg A) \cdot P(\neg A)), or : c = \frac = \frac 1 .


Alternative form

Another form of Bayes' theorem for two competing statements or hypotheses is: :P(A, B) = \frac. For an epistemological interpretation: For proposition ''A'' and evidence or background ''B'', * P(A) is the prior probability, the initial degree of belief in ''A''. * P(\neg A) is the corresponding initial degree of belief in ''not-A'', that ''A'' is false, where P(\neg A) =1-P(A) * P(B, A) is the conditional probability or likelihood, the degree of belief in ''B'' given that proposition ''A'' is true. * P(B, \neg A) is the conditional probability or likelihood, the degree of belief in ''B'' given that proposition ''A'' is false. * P(A, B) is the
posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
, the probability of ''A'' after taking into account ''B''.


Extended form

Often, for some partition of the sample space, the event space is given in terms of ''P''(''Aj'') and ''P''(''B'' ,  ''Aj''). It is then useful to compute ''P''(''B'') using the law of total probability: :P(B) = , :\Rightarrow P(A_i, B) = \frac\cdot In the special case where ''A'' is a binary variable: :P(A, B) = \frac\cdot


Random variables

Consider a sample space Ω generated by two random variables ''X'' and ''Y''. In principle, Bayes' theorem applies to the events ''A'' =  and ''B'' = . :P( Xx , Y y) = \frac However, terms become 0 at points where either variable has finite probability density. To remain useful, Bayes' theorem must be formulated in terms of the relevant densities (see Derivation).


Simple form

If ''X'' is continuous and ''Y'' is discrete, :f_(x) = \frac where each f is a density function. If ''X'' is discrete and ''Y'' is continuous, : P(Xx, Yy) = \frac. If both ''X'' and ''Y'' are continuous, : f_(x) = \frac.


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: : f_Y(y) = \int_^\infty f_(y) f_X(\xi)\,d\xi .


Bayes' rule in odds form

Bayes' theorem in odds form is: :O(A_1:A_2\mid B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2\mid B) where :\Lambda(A_1:A_2\mid B) = \frac is called the Bayes factor or likelihood ratio. The odds between two events is simply the ratio of the probabilities of the two events. Thus :O(A_1:A_2) = \frac, :O(A_1:A_2\mid B) = \frac, Thus, the rule says that the posterior odds are the prior odds times the Bayes factor, or in other words, the posterior is proportional to the prior times the likelihood. In the special case that A_1 = A and A_2 = \neg A, one writes O(A)=O(A:\neg A) =P(A)/(1-P(A)), and uses a similar abbreviation for the Bayes factor and for the conditional odds. The odds on A is by definition the odds for and against A. Bayes' rule can then be written in the abbreviated form :O(A\mid B) = O(A) \cdot \Lambda(A\mid B) , or, in words, the posterior odds on A equals the prior odds on A times the likelihood ratio for A given information B. In short, posterior odds equals prior odds times likelihood ratio. For example, if a medical test has a sensitivity of 90% and a specificity of 91%, then the positive Bayes factor is \Lambda_+ = P(\text)/P(\text) = 90\%/(100\%-91\%)=10. Now, if the prevalence of this disease is 9.09%, and if we take that as the prior probability, then the prior odds is about 1:10. So after receiving a positive test result, the posterior odds of actually having the disease becomes 1:1; In other words, the posterior probability of actually having the disease is 50%. If a second test is performed in serial testing, and that also turns out to be positive, then the posterior odds of actually having the disease becomes 10:1, which means a posterior probability of about 90.91%. The negative Bayes factor can be calculated to be 91%/(100%-90%)=9.1, so if the second test turns out to be negative, then the posterior odds of actually having the disease is 1:9.1, which means a posterior probability of about 9.9%. The example above can also be understood with more solid numbers: Assume the patient taking the test is from a group of 1000 people, where 91 of them actually have the disease (prevalence of 9.1%). If all these 1000 people take the medical test, 82 of those with the disease will get a true positive result (sensitivity of 90.1%), 9 of those with the disease will get a false negative result ( false negative rate of 9.9%), 827 of those without the disease will get a true negative result (specificity of 91.0%), and 82 of those without the disease will get a false positive result (false positive rate of 9.0%). Before taking any test, the patient's odds for having the disease is 91:909. After receiving a positive result, the patient's odds for having the disease is :\frac\times\frac=\frac=1:1 which is consistent with the fact that there are 82 true positives and 82 false positives in the group of 1000 people.


Correspondence to other mathematical frameworks


Propositional logic

Using P(\neg B\mid A)=1-P(B\mid A) twice, one may use Bayes' theorem to also express P(\neg B\mid \neg A) in terms of P(A\mid B) and without negations: :P(\neg B\mid \neg A) = 1 - \left(1-P(A\mid B)\right)\frac, when P(\neg A) = 1 - P(A) \neq 0. From this we can read off the inference :P(A\mid B) = 1\implies P(\neg B\mid \neg A) = 1. In words: If certainly B implies A, we infer that certainly \neg A implies \neg B. Where P(B)\neq 0, the two implications being certain are equivalent statements. In the probability formulas, the conditional probability P(A \mid B) generalizes the logical implication B \implies A, where now beyond assigning true or false, we assign probability values to statements. The assertion of B \implies A is captured by certainty of the conditional, the assertion of P(A\mid B) = 1. Relating the directions of implication, Bayes' theorem represents a generalization of the contraposition law, which in classical propositional logic can be expressed as: :(B \implies A)\iff(\neg A \implies \neg B). Note that in this relation between implications, the positions of A resp. B get flipped. The corresponding formula in terms of probability calculus is Bayes' theorem, which in its expanded form involving the prior probability/ base rate a of only A, is expressed as: :P(A \mid B) = P(B \mid A) \frac.


Subjective logic

Bayes' theorem represents a special case of deriving inverted conditional opinions in subjective logic expressed as: :(\omega^S_,\omega^S_) = (\omega^S_, \omega^S_) \widetilde a_A, where \widetilde denotes the operator for inverting conditional opinions. The argument (\omega^S_,\omega^S_) denotes a pair of binomial conditional opinions given by source S, and the argument a_ denotes the prior probability (aka. the base rate) of A. The pair of derivative inverted conditional opinions is denoted (\omega^S_,\omega^_). The conditional opinion \omega^S_ generalizes the probabilistic conditional P(A \mid B), i.e. in addition to assigning a probability the source S can assign any subjective opinion to the conditional statement (A\mid B). A binomial subjective opinion \omega^_ is the belief in the truth of statement A with degrees of epistemic uncertainty, as expressed by source S. Every subjective opinion has a corresponding projected probability P(\omega^_). The application of Bayes' theorem to projected probabilities of opinions is a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
, meaning that Bayes' theorem can be expressed in terms of projected probabilities of opinions: :P(\omega^S_) = \frac. Hence, the subjective Bayes' theorem represents a generalization of Bayes' theorem.


Generalizations


Conditioned version

A conditioned version of the Bayes' theorem results from the addition of a third event C on which all probabilities are conditioned: :P(A \mid B \cap C) = \frac


Derivation

Using the chain rule :P(A \cap B \cap C) = P(A \mid B \cap C) \, P(B \mid C) \, P(C) And, on the other hand :P(A \cap B \cap C) = P(B \cap A \cap C) = P(B \mid A \cap C) \, P(A \mid C) \, P(C) The desired result is obtained by identifying both expressions and solving for P(A \mid B \cap C).


Bayes' rule with 3 events

In the case of 3 events - A, B, and C - it can be shown that: P(A \mid B,C) = \frac


History

Bayes' theorem is named after the Reverend Thomas Bayes (; c. 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). He studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). On Bayes's death his family transferred his papers to his old friend, Richard Price (1723–1791) who over a period of two years significantly edited the unpublished manuscript, before sending it to a friend who read it aloud at the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
on 23 December 1763. 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 and chose one of the two solutions offered by Bayes. In 1765, Price was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes.Holland, pp. 46–7. On 27 April a letter sent to his friend
Benjamin Franklin Benjamin Franklin ( April 17, 1790) was an American polymath who was active as a writer, scientist, inventor, statesman, diplomat, printer, publisher, and political philosopher. Encyclopædia Britannica, Wood, 2021 Among the leading int ...
was read out at the Royal Society, and later published, where Price applies this work to population and computing 'life-annuities'.. Independently of Bayes,
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
in 1774, and later in his 1812 ''
Théorie analytique des probabilités Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
'', used conditional probability to formulate the relation of an updated
posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work. The Bayesian interpretation of probability was developed mainly by Laplace.
Sir Harold Jeffreys Sir Harold Jeffreys, FRS (22 April 1891 – 18 March 1989) was a British mathematician, statistician, geophysicist, and astronomer. His book, ''Theory of Probability'', which was first published in 1939, played an important role in the revival ...
put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". 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:


Use in genetics

In genetics, Bayes' theorem can be used to calculate the probability of an individual having a specific genotype. Many people seek to approximate their chances of being affected by a genetic disease or their likelihood of being a carrier for a recessive gene of interest. A Bayesian analysis can be done based on family history or genetic testing, in order to predict whether an individual will develop a disease or pass one on to their children. Genetic testing and prediction is a common practice among couples who plan to have children but are concerned that they may both be carriers for a disease, especially within communities with low genetic variance. The first step in Bayesian analysis for genetics is to propose mutually exclusive hypotheses: for a specific allele, an individual either is or is not a carrier. Next, four probabilities are calculated: Prior Probability (the likelihood of each hypothesis considering information such as family history or predictions based on Mendelian Inheritance), Conditional Probability (of a certain outcome), Joint Probability (product of the first two), and Posterior Probability (a weighted product calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities). This type of analysis can be done based purely on family history of a condition or in concert with genetic testing.


Using pedigree to calculate probabilities

Example of a Bayesian analysis table for a female individual's risk for a disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subject's siblings and parents, she is equally likely to be a carrier as to be a non-carrier (this likelihood is denoted by the Prior Hypothesis). However, the probability that the subject's four sons would all be unaffected is 1/16 (···) if she is a carrier, about 1 if she is a non-carrier (this is the Conditional Probability). The Joint Probability reconciles these two predictions by multiplying them together. The last line (the Posterior Probability) is calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities.


Using genetic test results

Parental genetic testing can detect around 90% of known disease alleles in parents that can lead to carrier or affected status in their child. Cystic fibrosis is a heritable disease caused by an autosomal recessive mutation on the CFTR gene, located on the q arm of chromosome 7."CFTR Gene – Genetics Home Reference". U.S. National Library of Medicine, National Institutes of Health, ghr.nlm.nih.gov/gene/CFTR#location. Bayesian analysis of a female patient with a family history of cystic fibrosis (CF), who has tested negative for CF, demonstrating how this method was used to determine her risk of having a child born with CF: Because the patient is unaffected, she is either homozygous for the wild-type allele, or heterozygous. To establish prior probabilities, a Punnett square is used, based on the knowledge that neither parent was affected by the disease but both could have been carriers: Given that the patient is unaffected, there are only three possibilities. Within these three, there are two scenarios in which the patient carries the mutant allele. Thus the prior probabilities are and . Next, the patient undergoes genetic testing and tests negative for cystic fibrosis. This test has a 90% detection rate, so the conditional probabilities of a negative test are 1/10 and 1.  Finally, the joint and posterior probabilities are calculated as before. After carrying out the same analysis on the patient's male partner (with a negative test result), the chances of their child being affected is equal to the product of the parents' respective posterior probabilities for being carriers times the chances that two carriers will produce an affected offspring ().


Genetic testing done in parallel with other risk factor identification.

Bayesian analysis can be done using phenotypic information associated with a genetic condition, and when combined with genetic testing this analysis becomes much more complicated. Cystic Fibrosis, for example, can be identified in a fetus through an ultrasound looking for an echogenic bowel, meaning one that appears brighter than normal on a scan2. This is not a foolproof test, as an echogenic bowel can be present in a  perfectly healthy fetus. Parental genetic testing is very influential in this case, where a phenotypic facet can be overly influential in probability calculation. In the case of a fetus with an echogenic bowel, with a mother who has been tested and is known to be a CF carrier, the posterior probability that the fetus actually has the disease is very high (0.64). However, once the father has tested negative for CF, the posterior probability drops significantly (to 0.16). Risk factor calculation is a powerful tool in genetic counseling and reproductive planning, but it cannot be treated as the only important factor to consider. As above, incomplete testing can yield falsely high probability of carrier status, and testing can be financially inaccessible or unfeasible when a parent is not present.


See also

*
Bayesian epistemology Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its conc ...
*
Inductive probability Inductive probability attempts to give the probability of future events based on past events. It is the basis for inductive reasoning, and gives the mathematical basis for learning and the perception of patterns. It is a source of knowledge about ...
* Quantum Bayesianism *'' Why Most Published Research Findings Are False'', a 2005 essay in
metascience Metascience (also known as meta-research) is the use of scientific methodology to study science itself. Metascience seeks to increase the quality of scientific research while reducing inefficiency. It is also known as "''research on research''" ...
by John Ioannidis


Notes


References


Bibliography

*


Further reading

* * Gelman, A, Carlin, JB, Stern, HS, and Rubin, DB (2003), "Bayesian Data Analysis," Second Edition, CRC Press. * Grinstead, CM and Snell, JL (1997), "Introduction to Probability (2nd edition)," American Mathematical Society (free pdf available

* * * * Lee, Peter M (2012), "Bayesian Statistics: An Introduction," 4th edition. Wiley. . * * Rosenthal, Jeffrey S (2005), "Struck by Lightning: The Curious World of Probabilities". HarperCollins. (Granta, 2008. ). * * Stone, JV (2013), download chapter 1 o
"Bayes' Rule: A Tutorial Introduction to Bayesian Analysis"
Sebtel Press, England. *
Bayesian Reasoning for Intelligent People
', An introduction and tutorial to the use of Bayes' theorem in statistics and cognitive science. * Morris, Dan (2016), Read first 6 chapters for free of
Bayes' Theorem Examples: A Visual Introduction For Beginners
Blue Windmill . A short tutorial on how to understand problem scenarios and find P(B), P(A), and P(B, A).


External links

* *

Contains origins of "Bayesian", "Bayes' Theorem", "Bayes Estimate/Risk/Solution", "Empirical Bayes", and "Bayes Factor".
A tutorial on probability and Bayes' theorem devised for Oxford University psychology students

An Intuitive Explanation of Bayes' Theorem by Eliezer S. Yudkowsky

Bayesian Clinical Diagnostic Model
{{DEFAULTSORT:Bayes' Theorem Bayesian statistics, Theorem Probability theorems Theorems in statistics