Bayes' theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search
A blue neon sign, showing the simple statement of Bayes's theorem

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) relates current to prior belief. It also relates current to prior evidence. It is important in the mathematical manipulation of conditional probabilities.[1] Bayes's rule can be derived from more basic axioms of probability, specifically conditional probability.

When applied, the probabilities involved in Bayes's theorem may have any of a number of probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. ln particular, with the Bayesian interpretation of probability, the theorem expresses how a subjective degree of belief should rationally change to account for evidence: this is Bayesian inference, which is fundamental to Bayesian statistics. However, Bayes's theorem has applications in a wide range of calculations involving probabilities, not just in Bayesian inference.

Bayes's theorem is named after Rev. Thomas Bayes (/ˈbz/; 1701–1761), who first showed how to use new evidence to update beliefs. Bayes' unpublished manuscript was significantly edited by Richard Price before it was posthumously read at the Royal Society. Bayes' algorithm remained unknown until it was independently rediscovered and further developed by Pierre-Simon Laplace, who first published the modern formulation in his 1812 Théorie analytique des probabilités.

Sir Harold Jeffreys put Bayes' algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes's theorem "is to the theory of probability what Pythagoras's theorem is to geometry".[2]

Introduction[edit]

Bayes's theorem is stated mathematically as the following simple form:[1]

P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}\cdot

For an epistemological interpretation:[1]

For proposition A and evidence or background B,

  • P(A), the prior probability, is the initial degree of belief in A.
  • P(A|B), the conditional probability, is the degree of belief in A having accounted for B.
  • the quotient P(B|A)/P(B) represents the support B provides for A.

Another form of Bayes's Theorem that is generally encountered when looking at two competing statements or hypotheses is:

P(A|B) = \frac{P(B|A)\,P(A)}{ P(B|A) P(A) + P(B|\neg A) P(\neg A)}\cdot

For an epistemological interpretation:

For proposition A and evidence or background B,[3]

  • P(A),the prior probability, is the initial degree of belief in A.
  • P(-A), is the corresponding probability of the initial degree of belief against A: 1-P(A)=P(-A)
  • P(B|A), the conditional probability or likelihood, is the degree of belief in B, given that the proposition A is true.
  • P(B|-A), 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.

Introductory example (epistemological interpretation)[edit]

Suppose a man told you he had a nice conversation with someone on the train. Not knowing anything about this conversation, the prior probability that he was speaking to a woman is 50% (assuming the speaker was as likely to strike up a conversation with a man as with a woman). Now suppose he also told you that his conversational partner had long hair. It is now more likely he was speaking to a woman, since women are more likely to have long hair than men. Bayes's theorem can be used to calculate the probability that the person was a woman.

To see how this is done, let W represent the event that the conversation was held with a woman, and L denote the event that the conversation was held with a long-haired person. It can be assumed that women constitute half the population for this example. So, not knowing anything else, the prior probability that W occurs is P(W) = 0.5.

Suppose it is also known that 75% of women have long hair, which we denote as P(L |W) = 0.75 (read: the probability of event L given event W is 0.75, meaning that the probability of a person having long hair (event "L"), given that we already know that the person is a woman ("event W") is 75%). This is the conditional probability that it was a woman who had the conversation on the train. Likewise, suppose it is known that 15% of men have long hair, or P(L |M) = 0.15, where M is the complementary event of W, i.e., the event that the conversation was held with a man (assuming that every human is either a man or a woman). This is the conditional probability against that it was a woman who had the conversation on the train.

Our goal is to calculate the posterior probability that the conversation was held with a woman, given the fact that the person had long hair, or, in our notation, P(W |L). Using the formula for Bayes's theorem, we have:

P(W|L) = \frac{P(L|W) P(W)}{P(L)} = \frac{P(L|W) P(W)}{P(L|W) P(W) + P(L|M) P(M)}

where we have used the law of total probability to expand P(L). The numeric answer can be obtained by substituting the above values into this formula (the algebraic multiplication is annotated using " · ", the centered dot). This yields

P(W|L) = \frac{0.75\cdot0.50}{0.75\cdot0.50 + 0.15\cdot0.50} = \frac56\approx 0.83,

i.e., the probability that the conversation was held with a woman, given that the person had long hair, is about 83%. More examples are provided below.

Another way to do this calculation is as follows. Initially, it is equally likely that the conversation is held with a woman as with a man, so the prior odds are 1:1. The respective chances that a man and a woman have long hair are 15% and 75%. It is 5 times more likely that a woman has long hair than that a man has long hair. We say that the likelihood ratio or Bayes factor is 5:1. Bayes's theorem in odds form, also known as Bayes's rule, tells us that the posterior odds that the person was a woman is also 5:1 (the prior odds, 1:1, times the likelihood ratio, 5:1). In a formula:

\frac{P(W|L)}{P(M|L)} = \frac{P(W)}{P(M)} \cdot \frac{P(L|W)}{P(L|M)}.

Statement and interpretation[edit]

Mathematically, Bayes's theorem gives the relationship between the probabilities of A and B, P(A) and P(B), and the conditional probabilities of A given B and B given A, P(A|B) and P(B|A). In its most common form, it is:

P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}\cdot

The meaning of this statement depends on the interpretation of probability ascribed to the terms:

Bayesian interpretation[edit]

Main article: Bayesian probability
A geometric visualisation of Bayes's theorem. In the table, the values ax, ay, bx and by give the relative weights of each corresponding condition and case. The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. This shows that P(A|X) P(X) = P(X|A) P(A) i.e. P(A|X) = P(X|A) P(A) / P(X). Similar reasoning can be used to show that P(B|X) = P(X|B) P(B) / P(X) etc.

In the Bayesian (or epistemological) interpretation, probability measures a degree of belief. Bayes's 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)/P(B) represents the support B provides for A.

For more on the application of Bayes's theorem under the Bayesian interpretation of probability, see Bayesian inference.

Frequentist interpretation[edit]

Illustration of frequentist interpretation with tree diagrams. Bayes's theorem connects conditional probabilities to their inverses.

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's 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's theorem serves as the link between these different partitionings.

Forms[edit]

Events[edit]

Simple form[edit]

For events A and B, provided that P(B) ≠ 0,

P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}\cdot \,

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's theorem then shows that the posterior probabilities are proportional to the numerator:

P(A|B) \propto P(A) \cdot P(B|A) \ (proportionality over A for given B).

In words: posterior is proportional to prior times likelihood (see Lee, 2012, Chapter 1).

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

P(A|B) = c \cdot P(A) \cdot P(B|A) \ and P(\neg A|B) = c \cdot P(\neg A) \cdot P(B|\neg A)\cdot

Adding these two formulas we deduce that

 c = \frac{1}{P(A) \cdot P(B|A) + P(\neg A) \cdot P(B|\neg A) } .

Extended form[edit]

Often, for some partition {Aj} of the event 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:

P(B) = {\sum_j P(B|A_j) P(A_j)},
\implies P(A_i|B) = \frac{P(B|A_i)\,P(A_i)}{\sum\limits_j P(B|A_j)\,P(A_j)}\cdot

In the special case where A is a binary variable:

P(A|B) = \frac{P(B|A)\,P(A)}{ P(B|A) P(A) + P(B|\neg A) P(\neg A)}\cdot

Random variables[edit]

Diagram illustrating the meaning of Bayes's theorem as applied to an event space generated by continuous random variables X and Y. Note that there exists an instance of Bayes's theorem for each point in the domain. In practice, these instances might be parametrized by writing the specified probability densities as a function of x and y.

Consider a sample space Ω generated by two random variables X and Y. In principle, Bayes's 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, Bayes's theorem may be formulated in terms of the relevant densities (see Derivation).

Simple form[edit]

If X is continuous and Y is discrete,

f_X(x|Y=y) = \frac{P(Y=y|X=x)\,f_X(x)}{P(Y=y)}.

If X is discrete and Y is continuous,

 P(X=x|Y=y) = \frac{f_Y(y|X=x)\,P(X=x)}{f_Y(y)}.

If both X and Y are continuous,

 f_X(x|Y=y) = \frac{f_Y(y|X=x)\,f_X(x)}{f_Y(y)}.

Extended form[edit]

Diagram illustrating how an event space generated by continuous random variables X and Y is often conceptualized.

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}^\infty f_Y(y|X=\xi )\,f_X(\xi)\,d\xi .

Bayes's rule[edit]

Main article: Bayes' rule

Bayes' rule is Bayes's theorem in odds form.

O(A_1:A_2|B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2|B)

where

\Lambda(A_1:A_2|B) = \frac{P(B|A_1)}{P(B|A_2)}

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

O(A_1:A_2) = \frac{P(A_1)}{P(A_2)},
O(A_1:A_2|B) = \frac{P(A_1|B)}{P(A_2|B)},

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[edit]

For events[edit]

Bayes's theorem may be derived from the definition of conditional probability:

P(A|B)=\frac{P(A \cap B)}{P(B)}, \text{ if } P(B) \neq 0, \!
P(B|A) = \frac{P(A \cap B)}{P(A)}, \text{ if } P(A) \neq 0, \!
\implies P(A \cap B) = P(A|B)\, P(B) = P(B|A)\, P(A), \!
\implies P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}, \text{ if } P(B) \neq 0.

For random variables[edit]

For two continuous random variables X and Y, Bayes's theorem may be analogously derived from the definition of conditional density:

f_X(x|Y=y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}
f_Y(y|X=x) = \frac{f_{X,Y}(x,y)}{f_X(x)}
\implies f_X(x|Y=y) = \frac{f_Y(y|X=x)\,f_X(x)}{f_Y(y)}.

Examples[edit]

Frequentist example[edit]

Tree diagram illustrating frequentist example. R, C, P and P bar are the events representing rare, common, pattern and no pattern. Percentages in parentheses are calculated. Note that three independent values are given, so it is possible to calculate the inverse tree (see figure above).

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's theorem (since any beetle can be only rare or common),

\begin{align}P(\text{Rare}|\text{Pattern}) &=
\frac{P(\text{Pattern}|\text{Rare})P(\text{Rare})} {P(\text{Pattern}|\text{Rare})P(\text{Rare}) \, + \, P(\text{Pattern}|\text{Common})P(\text{Common})} \\[8pt]
&= \frac{0.98 \times 0.001} {0.98 \times 0.001 + 0.05 \times 0.999} \\[8pt]
&\approx 1.9\%. \end{align}

Coin flip example[edit]

Concrete example from 5 August 2011 New York Times article by John Allen Paulos (quoted verbatim):

"Assume that you are presented with three coins, two of them fair and the other a counterfeit that always lands heads. If you randomly pick one of the three coins, the probability that it's the counterfeit is 1 in 3. This is the prior probability of the hypothesis that the coin is counterfeit. Now after picking the coin, you flip it three times and observe that it lands heads each time. Seeing this new evidence that your chosen coin has landed heads three times in a row, you want to know the revised posterior probability that it is the counterfeit. The answer to this question, found using Bayes's theorem (calculation mercifully omitted), is 4 in 5. You thus revise your probability estimate of the coin's being counterfeit upward from 1 in 3 to 4 in 5."

The calculation ("mercifully supplied") follows:


\begin{align}
P(\text{Biased coin}) &= \frac{1}{3} \\[8pt]
P(\text{Fair coin}) &= \frac{2}{3} \\[8pt]
P(\text{H}|\text{Fair coin}) &= \frac{1}{2} \\[8pt]
P(\text{HHH}|\text{Fair coin}) &= \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \\[8pt]
P(\text{HHH}|\text{Biased coin}) &= 1 \times 1 \times 1 = 1 \\[8pt]
P(\text{Biased coin}|\text{HHH}) &= \frac{P(\text{HHH}|\text{Biased coin})P(\text{Biased coin})}{P(\text{HHH}|\text{Biased coin})P(\text{Biased coin}) + P(\text{HHH}|\text{Fair coin})P(\text{Fair coin})} \\[8pt]
&= \frac{1 \times \frac{1}{3}}{1 \times \frac{1}{3} + \frac{1}{8} \times \frac{2}{3}} \quad = \quad \frac{\frac{1}{3}}{\frac{10}{24}} \quad = \quad \frac{4}{5} \\[8pt]
\end{align}

Drug testing[edit]

Tree diagram illustrating drug testing example. U, U bar, "+" and "−" are the events representing user, non-user, positive result and negative result. Percentages in parentheses are calculated.

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. If a randomly selected individual tests positive, what is the probability he or she is a user?


\begin{align}
P(\text{User}|\text{+}) &= \frac{P(\text{+}|\text{User}) P(\text{User})}{P(\text{+}|\text{User}) P(\text{User}) + P(\text{+}|\text{Non-user}) P(\text{Non-user})} \\[8pt]
&= \frac{0.99 \times 0.005}{0.99 \times 0.005 + 0.01 \times 0.995} \\[8pt]
&\approx 33.2\%
\end{align}

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 (0.995%) outweighs the number of true positives (0.495%). 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 × 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, about 33%, are genuine.

Note: The importance of specificity can be illustrated by showing that even if sensitivity is 100% and specificity is at 99% the probability of the person being a drug user is ≈33% but if the specificity is changed to 99.5% and the sensitivity is dropped down to 99% the probability of the person being a drug user rises to 49.8%.

Applications[edit]

Bayes's theorem is significantly important in inverse problem theory, where the a posteriori probability density function is obtained from the product of prior probability density function and the likelihood probability density function. An important application is constructing computational models of oil reservoirs given the observed data.[4]

Although Bayes's theorem is commonly used to determine the probability of an event occurring, it can also be applied to verify someone's credibility as a prognosticator. Many pundits claim to be able to predict the outcome of an event; political elections, trials, the weather and even sporting events. Larry Sabato, founder of Sabato's Crystal Ball, is a perfect example. His website provides free political analysis and election predictions. His success at predictions has even led him to be called a "pundit with an opinion for every reporter's phone call."[5] We even have Punxsutawney Phil, the famous groundhog, who tells us whether or not we can expect a longer winter or an early spring. Bayes's theorem tells us the difference between who's on a hot streak and who is what they claim to be.

Let's say we live in an area where everyone gambles on the outcome of coin flips. Because of that, there is a big business for predicting coin flips. Suppose that 5% of predictors can actually win in the long run, and 80% of those are winners over a 2-year period. 95% of predictors are pretenders who are just guessing, and 20% of them are winners over a 2-year period (everyone gets lucky once in a while). This means that 82.6% of bettors who are winners over a 2-year period are actually long-term losers who are winning above their real average.

History[edit]

Bayes's theorem was named after the Reverend Thomas Bayes (1701–61), who studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). His friend Richard Price edited and presented this work in 1763, after Bayes's death, as An Essay towards solving a Problem in the Doctrine of Chances.[6] The French mathematician Pierre-Simon Laplace reproduced and extended Bayes's results in 1774, apparently quite unaware of Bayes's work.[7][8] Stephen Stigler suggested in 1983 that Bayes's theorem was discovered by Nicholas Saunderson some time before Bayes.[9] However, this interpretation has been disputed.[10]

Martyn Hooper[11] and Sharon McGrayne[12] have argued that Richard Price's contribution was substantial:

By modern standards, we should refer to the Bayes–Price rule. Price discovered Bayes's work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes's name alone is unfair but so entrenched that anything else makes little sense.

[12]

See also[edit]

Notes[edit]

  1. ^ a b c "Computational Statistics". University of Utah College of Engineering. "Bayes’ theorem (also known as Bayes’ rule or Bayes’ law) is a result in probability theory that relates conditional probabilities. If A and B denote two events, P(A/B) denotes the conditional probability of A occurring, given that B occurs. The two conditional probabilities P(A/B) and P(B/A) are in general different. Bayes theorem gives a relation between P(A/B) and P(B/A)." 
  2. ^ Jeffreys, Harold (1973). Scientific Inference (3rd ed.). Cambridge University Press. p. 31. ISBN 978-0-521-18078-8. 
  3. ^ "Bayes Theorem: Introduction". Trinity University. 
  4. ^ Gharib Shirangi, M., History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm, Journal of Petroleum Science and Engineering, http://www.sciencedirect.com/science/article/pii/S0920410513003227
  5. ^ Perry, James M. (July 18, 1994). "Sabato, 'Dr. Dial-a-Quote' of Political Scientists, Dispenses Advice to Candidates, Spin to the Press". Wall Street Journal. p. A14. 
  6. ^ Bayes, Thomas, and 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.". Philosophical Transactions of the Royal Society of London 53 (0): 370–418. doi:10.1098/rstl.1763.0053. 
  7. ^ Laplace refined Bayes's theorem over a period of decades:
    • Laplace announced his independent discovery of Bayes's 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's theorem appears on p. 29.
    • Laplace presented a refinement of Bayes's 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's 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.
  8. ^ Daston, Lorraine (1988). Classical Probability in the Enlightenment. Princeton Univ Press. p. 268. ISBN 0-691-08497-1. 
  9. ^ Stigler, Stephen M. (1983), "Who Discovered Bayes' Theorem?", The American Statistician 37(4):290–296. doi:10.1080/00031305.1983.10483122
  10. ^ Edwards, A. W. F. (1986), "Is the Reference in Hartley (1749) to Bayesian Inference?", The American Statistician 40(2):109–110 doi:10.1080/00031305.1986.10475370
  11. ^ Hooper, Martyn. (2013), "Richard Price, Bayes' theorem, and God", Significance 10(1):36–39. doi:10.1111/j.1740-9713.2013.00638.x
  12. ^ a b McGrayne, S. B. (2011). The Theory That Would Not Die: How Bayes' 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. 

Further reading[edit]

  • Bruss, F. Thomas (2013), "250 years of '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.' ", 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), "Bayesian Data Analysis", Second Edition, CRC Press.
  • Grinstead, CM and Snell, JL (1997), "Introduction to Probability (2nd edition)", American Mathematical Society (free pdf available) [1].
  • Hazewinkel, Michiel, ed. (2001), "Bayes formula", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  • McGrayne, SB (2011). The Theory That Would Not Die: How Bayes' 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), "Memoir on the Probability of the Causes of Events", Statistical Science 1(3):364–378.
  • Lee, PM (2012), "Bayesian Statistics: An Introduction", Wiley.
  • Rosenthal, JS (2005), "Struck by Lightning: the Curious World of Probabilities". Harper Collings.
  • Stigler, SM (1986), "Laplace's 1774 Memoir on Inverse Probability", Statistical Science 1(3):359–363.
  • Stone, JV (2013), download chapter 1 of "Bayes' Rule: A Tutorial Introduction to Bayesian Analysis", Sebtel Press, England.

External links[edit]