R: Bayes' Theorem

BayesTheorem {LaplacesDemon} | R Documentation

## Bayes' Theorem

### Description

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.

### Usage

    BayesTheorem(PrA, PrBA)

### Arguments

`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.

### Details

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.

### Value

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`.

### Author(s)

Statisticat, LLC.

### References

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.

### See Also

`IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `PMC`, and `VariationalBayes`.

### Examples

    # Pr(Hell|Consort) =
    PrA <- c(0.75,0.25)
    PrBA <- c(6/9, 5/7)
    BayesTheorem(PrA, PrBA)

[Package LaplacesDemon version 16.1.1 Index]