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) = \frac{\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(\mathrm{Hell} | \mathrm{Consort}) = \frac{\Pr(\mathrm{Consort} | \mathrm{Hell})\Pr(\mathrm{Hell})}{\Pr(\mathrm{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) = \frac{\Pr(B | A_i)\Pr(A_i)}{\Pr(B | A_i)\Pr(A_i) +\dots+ \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(\mathrm{Hell})

• \Pr(A_2) = \Pr(\mathrm{Heaven})

• \Pr(B) = \Pr(\mathrm{Consort})

• \Pr(A_1 | B) = \Pr(\mathrm{Hell} | \mathrm{Consort})

• \Pr(A_2 | B) = \Pr(\mathrm{Heaven} | \mathrm{Consort})

• \Pr(B | A_1) = \Pr(\mathrm{Consort} | \mathrm{Hell})

• \Pr(B | A_2) = \Pr(\mathrm{Consort} | \mathrm{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(\mathrm{Consort} | \mathrm{Hell}) = 6/9 = 0.666

• \Pr(\mathrm{Consort} | \mathrm{Heaven}) = 5/7 = 0.714

• \Pr(\mathrm{Hell}) = 0.75

• \Pr(\mathrm{Heaven}) = 0.25

Finally, the desired conditional probability \Pr(\mathrm{Hell} | \mathrm{Consort}) is calculated using Bayes' theorem

• \Pr(\mathrm{Hell} | \mathrm{Consort}) = \frac{0.666(0.75)}{0.666(0.75) + 0.714(0.25)}

• \Pr(\mathrm{Hell} | \mathrm{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.

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 16.1.6 Index]