Context: insurance, actuarial science, Bayesian probability

Credibility theory is a method by which the premium charged by an insurance company to a policyholder (or a group of policyholders) can be adjusted, given the new information on the policyholder's experience.

Perhaps an example will explain this better. Suppose you are about to purchase fire insurance for your building. How does the insurance company work out the approriate premium? Since it does not have a lot of information about you or the building, so the best it can do is to assume that you are an "average" policyholder and the building is an "average" building, and charge you the premium suitable for the average building. (We shall ignore the effects of underwriting.)

Suppose now that for some reason or another, you have not made a claim for, say, five years. Does this mean that you are a better risk for the insurance company (so that it can charge you a lower premium), or did your experience (of not claiming for five years) only a result of randomness? (Again, we need to ignore no claim discounts here.) Credibility theory allows the insurance company to assign a "credibility factor" on your past experience, so that your past experience plays a part in calculating the future premium (or the "credibility premium").

The simplest way of calculating the credibility premium P is:

P = ZX + (1 - Z)Y

where X and Y are premium estimates from two sources (e.g. one policyholder's past experience and the premium estimate for the whole group of policyholders), and Z is the credibility factor of X.

The actual calculation of the credibility premium depends on the type of information you have, but are all centred around Bayesian probability. Technically speaking, it involves finding the expected value of the posterior distribution of some parameter(s) given the prior distribution of the parameter(s) and some collateral information.