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Derivation in the discrete case

To derive Bayes' théorem in the discrete case, note first from the definition of conditional probability that

denoting by P(A,B) the joint probability of A and B.

Dividing the left- and right-hand sides by P(B), we obtain

éach term in Bayes' théorem has a conventional name.

The term P(A) is called the prior probability of A.

It is "prior" in the sense that it precedes any information about B.

P(A) is also the marginal probability of A.

The term P(A|B) is called the posterior probability of A, given B. It is "posterior" in the sense that it is derived from or entailed by the specified value of B.

The term P(B|A), for a specific value of B, is called the likelihood function for A given B and can also be written as L(A|B).

The term P(B) is the prior or marginal probability of B, and acts as the normalizing constant.

Alternative forms of Bayes' theorem

Bayes' théorem is often embellished by noting that

so the théorem can be restated as

where AC is the complementary event of A. More generally, where {Ai} forms a partition of the event space,

for any Ai in the partition.

See also the law of total probability.

Bayes' theorem for probability densities

There is also a version of Bayes' théorem for continuous distributions.

It is somewhat harder to derive, since probability densities,

strictly spéaking, are not probabilities,

so Bayes' théorem has to be established by a limit process;

see Papoulis (citation below), Section 7.3 for an elementary derivation.

Bayes' théorem for probability densities is formally similar to the théorem for probabilities:

and there is an analogous statement of the law of total probability:

As in the discrete case,

the terms have standard names.

f(x, y) is the joint distribution of X and Y,

f(x|y) is the posterior distribution of X given Y=y,

f(y|x) = L(x|y) is (as a function of x) the likelihood function of X given Y=y,

and f(x) and f(y) are the marginal distributions of X and Y respectively, with f(x) being the prior distribution of X.

Here we have indulged in a conventional abuse of notation,

using f for éach one of these terms,

although éach one is réally a different function;

the functions are distinguished by the names of their arguments.