  in bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence or background is taken into account. similarly, the posterior probability distribution is the probability distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained from an experiment or survey. "posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined. ==definition== the posterior probability is the probability of the parameters \theta given the evidence x: p(\theta|x). it contrasts with the likelihood function, which is the probability of the evidence given the parameters: p(x|\theta). the two are related as follows: 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)}. the posterior probability can be written in the memorable form as :\text{posterior probability} \propto \text{likelihood} \times \text{prior probability}. ==example== 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. 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 [[la
Bayesian_statistics