Bayesian Statistics

  An important school of statistical theory, in which statistics is derived from a probability interpretation that includes the degree of belief in a hypothesis. It thus refers not only to repeatable measurements (as does the frequentist interpretation). The interpretation of data can be described by Bayes Theorem:

where H is a hypothesis and d is experimental data. The Bayesian meaning of the different terms is:

• - P(H|d) is the degree of belief in the hypothesis H, after the experiment which produced data d.
• - P(H) is the prior probability of H being true.
• - P(d|H) is the ordinary likelihood function used also by non-Bayesians.
• - P(d) is the prior probability of obtaining data d. It can be rewritten using the other terms as: , where summation runs over all hypotheses.

What is called a ``Bayesian'' viewpoint is the application of the laws of probability to non-repeatable events: H is a hypothesis or proposition, either true or untrue, and P(H) is interpreted as the degree of belief in the proposition. For further discussion, see [Eadie71], [Press95], [Sivia96].