| 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 |
PrBA |
This required argument is the conditional probability of
|
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.
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)