Bayes' Theorem
Bayes' theorem shows the relation between two conditional probabilities that are the reverse of each other. This theorem is named after Reverend Thomas Bayes (1702-1761), and is also referred to as Bayes' law or Bayes' rule (Bayes and Price, 1763). Bayes' theorem expresses the conditional probability, or ‘posterior probability’, of an event A after B is observed in terms of the 'prior probability' of A, prior probability of B, and the conditional probability of B given A. Bayes' theorem is valid in all common interpretations of probability. This function provides one of several forms of calculations that are possible with Bayes' theorem.
BayesTheorem(PrA, PrBA)
PrA |
This required argument is the prior probability of A, or Pr(A). |
PrBA |
This required argument is the conditional probability of B given A or Pr(B | A), and is known as the data, evidence, or likelihood. |
Bayes' theorem provides an expression for the conditional probability of A given B, which is equal to
Pr(A | B) = (Pr(B | A)Pr(A)) / Pr(B)
For example, suppose one asks the question: what is the probability of going to Hell, conditional on consorting (or given that a person consorts) with Laplace's Demon. By replacing A with Hell and B with Consort, the question becomes
Pr(Hell | Consort) = (Pr(Consort | Hell)Pr(Hell)) / Pr(Consort)
Note that a common fallacy is to assume that Pr(A | B) = Pr(B | A), which is called the conditional probability fallacy.
Another way to state Bayes' theorem (and this is the form in the provided function) is
Pr(A[i] | B) = (Pr(B | A[i])Pr(A[i])) / (Pr(B | A[i])Pr(A[i]) +...+ Pr(B | A[n])Pr(A[n]))
Let's examine our burning question, by replacing A[i] with Hell or Heaven, and replacing B with Consort
Pr(A[1] = Pr(Hell)
Pr(A[2] = Pr(Heaven)
Pr(B) = Pr(Consort)
Pr(A[1] | B) = Pr(Hell | Consort)
Pr(A[2] | B) = Pr(Heaven | Consort)
Pr(B | A[1]) = Pr(Consort | Heaven)
Pr(B | A[2]) = Pr(Consort | Heaven)
Laplace's Demon was conjured and asked for some data. He was glad to oblige.
6 people consorted out of 9 who went to Hell.
5 people consorted out of 7 who went to Heaven.
75% of the population goes to Hell.
25% of the population goes to Heaven.
Now, Bayes' theorem is applied to the data. Four pieces are worked out as follows
Pr(Consort | Hell) = 6/9 = 0.666
Pr(Consort | Heaven) = 5/7 = 0.714
Pr(Hell) = 0.75
Pr(Heaven) = 0.25
Finally, the desired conditional probability Pr(Hell | Consort) is calculated using Bayes' theorem
Pr(Hell | Consort) = 0.666(0.75) / (0.666(0.75) + 0.714(0.25))
Pr(Hell | Consort) = 0.737
The probability of someone consorting with Laplace's Demon and going to Hell is 73.7%, which is less than the prevalence of 75% in the population. According to these findings, consorting with Laplace's Demon does not increase the probability of going to Hell.
For an introduction to model-based Bayesian inference, see the accompanying vignette entitled “Bayesian Inference” or https://web.archive.org/web/20150206004608/http://www.bayesian-inference.com/bayesian.
The BayesTheorem function returns the conditional probability
of A given B, known in Bayesian inference as the
posterior. The returned object is of class bayestheorem.
Statisticat, LLC.
Bayes, T. and Price, R. (1763). "An Essay Towards Solving a Problem in the Doctrine of Chances". By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, M.A. and F.R.S. Philosophical Transactions of the Royal Statistical Society of London, 53, p. 370–418.
# Pr(Hell|Consort) = PrA <- c(0.75,0.25) PrBA <- c(6/9, 5/7) BayesTheorem(PrA, PrBA)
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