**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.