All About Programming: Posterior probability - Wikipedia, the free encyclopedia

Posterior probability - Wikipedia, the free encyclopedia

Suppose there is a mixed school having 60% boys and 40% girls as students. The girls 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.

Let us have a prior belief that the probability distribution function is

p(\theta)

and observations

x

with the likelihood

p(x|\theta)

, then the posterior probability is defined as

p(\theta|x) = \frac{p(x|\theta)p(\theta)}{p(x)}.

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The posterior probability can be written in the memorable form as

\text{Posterior probability} \propto \text{Likelihood} \times \text{Prior probability}

The event

G

is that the student observed is a girl, and the event

T

is that the student observed is wearing trousers. To compute

P(G|T)

, we first need to know:

$P(G)$ , or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4.
$P(B)$ , or the probability that the student is not a girl (i.e. a boy) regardless of any other information ( $B$ is the complementary event to $G$ ). This is 60%, or 0.6.
$P(T|G)$ , 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.
$P(T|B)$ , or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
$P(T)$ , or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since $P(T) = P(T|G)P(G) + P(T|B)P(B)$ (via the law of total probability), this is $P(T)= 0.5\times0.4 + 1\times0.6 = 0.8$ .

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(G|T) = \frac{P(T|G) P(G)}{P(T)} = \frac{0.5 \times 0.4}{0.8} = 0.25.

Read full article from Posterior probability - Wikipedia, the free encyclopedia

Posterior probability - Wikipedia, the free encyclopedia

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