User:Bjcairns/scratch

This page exists for me to experiment with and draft text for Wikipedia. If you came here by mistake, you may want to head back to my main user page. Ben Cairns 23:02, 16 Nov 2003 (UTC)

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Bayes' theorem in Bayesian inference

The example above demonstrates how the probability that a statement is true, in that case that a patient has a particular disease, can be improved by testing. Initially, the only information available is the incidence of the disease in the general population, but after testing the probability that the patient had the disease was refined by conditioning on the test results.

In Bayesian inference, we may substitute the word 'statement' with 'hypothesis' and the words 'test results' with 'data'. Again in the context of the above example, we might hypothesise that the patient has the disease, and the outcome of the test provides us with data that we can use to find the probability that our hypothesis is true given the data.

This is an uncontroversial example: neither frequentists nor Bayesians should object to the application of Bayes' theorem to such cases, since the hypothesis in question does have a frequentist interpretation, and the probabilities may easily be regarded by Bayesians as 'degrees-of-belief'. In Bayesian inference, however, hypotheses are permissable that do not permit frequentist interpretations.

For example, no frequentist interpretation exists for the probability that there was life on Mars 1 billion years ago, because there is no population of 'Marses' over which to define the probability in terms of the frequency of the event. (In contrast, frequentists might ask, what is the probability that there was life on any randomly-chosen planet in the universe 1 billion years ago.) Bayesians, however, defining probability as a degree-of-belief, may assign this hypothesis a probability based on their belief about its truth or falsity, prior to any investigation into the matter. This is called the prior probability.