Bayes' Theorem

Let  and  be sets. Conditional probability requires that

(1)

where  denotes intersection ("and"), and also that

(2)

Therefore,

(3)

Now, let

(4)

so  is an event in  and  for , then

(5)

(6)

But this can be written

(7)

so

(8)

(Papoulis 1984, pp. 38-39).

RFERENCES:

Papoulis, A. "Bayes' Theorem in Statistics" and "Bayes' Theorem in Statistics (Reexamined)." §3-5 and 4-4 in Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 38-39, 78-81, and 112-114, 1984.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 810, 1992.