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Prior probability

by Marco Taboga, PhD

The prior probability is one of the quantities involved in Bayes' rule:[eq1]It is the probability [eq2] assigned to the event A before receiving the information that the event $B$ has happened. After receiving this information, the prior probability is updated and the posterior probability [eq3] is computed, exploiting the knowledge of the conditional probability [eq4].

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Suppose that an individual is extracted at random from a population consisting of two ethnic groups, ABC and XYZ. We know that 30% of individuals belonging to group ABC have incomes below the poverty line, while the corresponding proportion for the population as a whole is 20%. Furthermore, 40% of the population is made of individuals belonging to group ABC. If we extract an individual whose income is below the poverty line, what is the probability that she belongs to group ABC?

This conditional probability can be computed by using Bayes' rule. The quantities involved in the computation are[eq5]

The prior probability is [eq6], the probability of belonging to group ABC, while the posterior probability [eq7] can be computed thanks to Bayes' rule:[eq8]

More details

You can find a more exhaustive explanation of the concept of prior probability in the lecture entitled Bayes' rule.

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