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In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability given the relevant evidence or background. "Posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined.
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.
The posterior probability is the probability of the parameters given the evidence : .
It contrasts with the likelihood function, which is the probability of the evidence given the parameters: .
The two are related as follows:
Given a prior belief that a probability distribution function is and that the observations have a likelihood , then the posterior probability is defined as
where is the normalizing constant and is calculated as
for continuous , or by summing over all possible values of for discrete .
The posterior probability is therefore proportional to the product Likelihood · Prior probability.
Suppose there is a school having 60% boys and 40% girls as students. The girls wear trousers or skirts in equal numbers; all boys 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 is that the student observed is a girl, and the event is that the student observed is wearing trousers. To compute the posterior probability , we first need to know:
Given all this information, the posterior 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:
An intuitive way to solve this is to assume the school has N students. Number of boys = 0.6N and number of girls = 0.4N. If N is sufficiently large, total number of trouser wearers = 0.6N+ 50% of 0.4N. And number of girl trouser wearers = 50% of 0.4N. Therefore, in the population of trousers, girls are (50% of 0.4N)/(0.6N+ 50% of 0.4N) = 25%. In other words, if you separated out the group of trouser wearers, a quarter of that group will be girls. Therefore, if you see trousers, the most you can deduce is that you are looking at a single sample from a subset of students where 25% are girls. And by definition, chance of this random student being a girl is 25%. Every Bayes theorem problem can be solved in this way.
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:
gives the posterior probability density function for a random variable given the data , where
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