Posterior probability: Definition from Answers.com

In Bayesian statistics posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence is taken into account.

1 Definition
2 Example
3 Calculation
4 References
5 See also

Definition

Let us have a priori belief that the probability distribution function is $p (θ)$ and an observation $X$ with the likelihood $p (X | θ)$ , then the posterior probability is defined as $p(\theta|X) \propto p(\theta)p(X|\theta)$ . The posterior probability can be written in the memorable form as $\mbox{Posterior probability} \propto \mbox{Prior probability} \times \mbox{Likelihood}$ .

Example

Suppose there is a co-ed school having 60% boys and 40% girls as students. The girl students wear trousers or skirts in equal numbers; the boys all wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem.

The event A is that the student observed is a girl, and the event B is that the student observed is wearing trousers. To compute P(A|B), we first need to know:

P(A), or the probability that the student is a girl regardless of any other information. Since the observers sees a random student, meaning that all students have the same probability of being observed, and the fraction of girls among the students is 40%, this probability equals 0.4.
P(A'), or the probability that the student is a boy regardless of any other information (A' is the complementary event to A). This is 60%, or 0.6.
P(B|A), or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.
P(B|A'), or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
P(B), or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since P(B) = P(B|A)P(A) + P(B|A')P(A'), this is 0.5×0.4 + 1×0.6 = 0.8.

Given all this information, the probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

$P(A|B) = \frac{P(B|A) P(A)}{P(B)} = \frac{0.5 \times 0.4}{0.8} = 0.25.$

Calculation

The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:

$f_{X\mid Y=y}(x)={f_X(x) L_{X\mid Y=y}(x) \over {\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx}}$

gives the posterior probability density function for a random variable X given the data Y = y, where

$f X (x)$ is the prior density of X,

$L_{X\mid Y=y}(x) = f_{Y\mid X=x}(y)$ is the likelihood function as a function of x,

$\int_{-\infty}^\infty f_X(x) L_{X\mid Y=y}(x)\,dx$ is the normalizing constant, and

$f_{X\mid Y=y}(x)$ is the posterior density of X given the data Y = y.

References

Peter M. Lee (2004). Bayesian Statistics, an introduction (3rd ed.). Wiley. ISBN 978-0-340-81405-5.

	Bayes's theorem (philosophy)
	prior probability (philosophy)
	odds

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Posterior probability

Contents

Definition

Example

Calculation

References

See also