Description Usage Arguments Details Value Author(s) References See Also Examples
Bayes' theorem shows the relation between two conditional probabilities that are the reverse of each other. This theorem is named after Reverend Thomas Bayes (17021761), 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.
1  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 modelbased Bayesian inference, see the accompanying vignette entitled “Bayesian Inference” or https://web.archive.org/web/20150206004608/http://www.bayesianinference.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.
IterativeQuadrature
,
LaplaceApproximation
,
LaplacesDemon
,
PMC
, and
VariationalBayes
.
1 2 3 4  # Pr(HellConsort) =
PrA < c(0.75,0.25)
PrBA < c(6/9, 5/7)
BayesTheorem(PrA, PrBA)

[1] 0.7368421 0.2631579
attr(,"class")
[1] "bayestheorem"
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