Bayes' theorem is a result in probability theory, named after the Reverend Thomas Bayes, who proved a special case of it in the 18th century. It is used in statistical inference to update estimates of the probability that different hypotheses are true, based on observations and a knowledge of how likely those observations are, given each hypothesis. Its discrete version may appear to go little beyond an identity that is sometimes taken to be the definition of conditional probability, but there is also a continuous version. A frequent error is to think that reliance on Bayes' theorem is the essence of Bayesianism, whose essence is actually the degreeofbelief interpretation of probability, contrasted with various "frequency" interpretations.
We will start with the simplest case of only two hypotheses, H_{1} and H_{2}. Suppose that we know that precisely one of the two hypotheses must be true, and suppose furthermore that we know their "prior" probabilities P(H_{1}) and P(H_{2}) = 1  P(H_{1}). Now some "data" D is observed, and we know the conditional probabilities of D given H_{1} and H_{2}, written as P(D  H_{1}) and P(D  H_{2}). We want to compute the "posterior" probabilities of H_{1} and H_{2}, given the observation of D. Bayes' theorem states that these probabilities can be computed as
To illustrate, suppose there are two bowls full of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Somebody randomly picks a bowl, and then randomly picks a cookie. The cookie turns out to be a plain one. How likely is it that he picked it out of bowl #1?
Intuitively, it seems clear that the answer should be more than 50%, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. H_{1} corresponds to bowl #1, and H_{2} to bowl #2. Since the bowl was picked randomly, we know P(H_{1}) = P(H_{2}) = 50%. The "data" D consists in the observation of a plain cookie. From the contents of the bowls, we know that P(D  H_{1}) = 75% and P(D  H_{2}) = 50%. Bayes' formula then yields
The theorem is also true if we have more than just two hypotheses, say H_{1}, H_{2}, H_{3}, ..., of which precisely one is true. Suppose we know the prior probability distribution
The continuous case of Bayes' theorem also says the posterior distribution results from multiplying the prior by the likelihood and then normalizing. The prior and posterior distributions are usually identified with their probability density functions.
For example, suppose the proportion of voters who will vote "yes" is an unknown number p between 0 and 1. A sample of n voters is drawn randomly from the population, and it is observed that x of those n voters will vote "yes". The likelihood function is then
Bayesianism is the philosophical tenet that the rules of mathematical probability apply not only when probabilities are relative frequencies assigned to random events, but also when they are degrees of belief assigned to uncertain propositions. Updating these degrees of belief in light of new evidence almost invariably involves application of Bayes' theorem.
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