BayesTheorem: 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.

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.

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.

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.

Examples

Run this code

# NOT RUN {
# Pr(Hell|Consort) =
PrA <- c(0.75,0.25)
PrBA <- c(6/9, 5/7)
BayesTheorem(PrA, PrBA)
# }

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