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Box 17.2Use of Bayes' theorem for combining probabilities

A formal statement of Bayes' theorem is:

P(Hi) means the probability of the ith hypothesis, and the vertical line means ‘given’, so that P(E | Hi) means the probability of the evidence (E) given hypothesis Hi.

The steps in performing a Bayesian calculation are as follows:

1. Set up a table with one column for each of the alternative hypotheses. Cover all the alternatives.
2. Assign a prior probability to each alternative. The prior probabilities of all the hypotheses must sum to 1. It is not important at this stage to worry about exactly what information you should use to decide the prior probability, as long as it is consistent across the columns. You will not be using all the information (otherwise there would be no point in doing the calculation because you would already have the answer) and any information not used in the prior probability can be used later.
3. Using one item of information not included in the prior probabilities, calculate a conditional probability for each hypothesis. The conditional probability is the probability of the information, given the hypothesis P(E | Hi) [not the probability of the hypothesis given the information, P(Hi | E)]. The conditional probabilities for the different hypotheses do not necessarily sum to 1.
4. If there are further items of information not yet included, repeat step (iii) as many times as necessary until all information has been used once and once only. The end result is a number of lines of conditional probabilities in each column.
5. Within each column, multiply together the prior and all the conditional probabilities. This gives a joint probability. The joint probabilities do not necessarily sum to 1 across the columns.
6. If there are just two columns, the joint probabilities can be used directly as odds. Alternatively the joint probabilities can be scaled to give final probabilities which do sum to 1. This is done by dividing each joint probability by the sum of all the joint probabilities.
Human Molecular Genetics. 2nd edition.