98
edits
Changes
Jump to navigation
Jump to search
New page: Bayes' theorem is a result in probability theory that shows how to invert a conditional probability. An extended form of the theorem provides a rigorous mathematical way of learnin...
Bayes' theorem is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous mathematical way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].
Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus
p(AB) = p(A)p(B|A) = p(B)p(A|B)
This can be rearranged to be
p(B|A) = p(B)p(A|B)/p(A)
This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-19761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].
Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus
p(AB) = p(A)p(B|A) = p(B)p(A|B)
This can be rearranged to be
p(B|A) = p(B)p(A|B)/p(A)
This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-19761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].