Prior Probability

The a priori distribution is a term used in Bayesian statistics.

Definition

The following situation is given: is an unknown population parameter to be estimated on the basis of observations of a random variable.

Given a distribution for the parameters, which describes the knowledge of the parameters prior to the observation of the sample. This distribution is known as the a priori distribution.

Furthermore, the conditional distribution of the sample is given under the condition which is also known as a likelihood function.

From the a priori distribution and the likelihood function can be calculated by means of the theorem of Bayes, the a posteriori distribution, which is essential for the calculation of point estimates (see Bayes estimator ) and interval estimators in Bayesian statistics.

(Non) informative a priori distributions

A non-informative a priori distribution is defined as an a priori distribution, which has no influence on the a posteriori distribution. This provides an a posteriori distribution which is identical with the likelihood function. Maximum a posteriori estimates and confidence intervals, which were obtained with a non- informative a priori distribution are therefore numerically equivalent to maximum likelihood estimators and frequentist confidence intervals.

An informative a priori distribution is present in all other cases.

The concept of non- informative a priori distribution is explained by an example: The random variable Y is the average intelligence quotient in the city ZZZ. Due to the design of the IQs is known that Y is normally distributed with standard deviation of 15 and an unknown parameter. In a sample of N volunteers of the intelligence quotient is measured. In this sample, an arithmetic mean of 105 is observed.

A A non-informative prior distribution is given in this case by

Wherein a positive real number. In this way obtained as a posterior distribution of a normal distribution with mean and standard deviation of 105. The maximum a posteriori estimator for the mean is then 105 (ie: the arithmetic mean of the sample) and thus identical to the maximum likelihood estimator.

Propere vs. Impropere a priori distributions

In the example above, a problem can be illustrated that often occurs with the use of non- informative a priori distributions: defining a so-called impropere a priori distribution. Impropere a priori distributions are characterized in that the integral of the a priori distribution is greater than 1. Therefore impropere a priori distributions are no probability distributions. In many cases, however, it can be shown that the a posteriori distribution is also defined using a distribution improperen. This is true when

Applies to all. A propere A prior distribution is defined by the fact that it is independent of the data and that their integral gives the value 1.

A conjugate prior distributions

A conjugate a priori distribution is always present if the a priori distribution is of the same type as the distribution of the a posteriori distribution.

An example is the binomial Beta Model: X is a binomial random variable with probability of success p as a parameter. In individual experiments N k successes are observed. As the a priori distribution of p beta (a, b ) distribution is used. Under these conditions, the posterior distribution is a Beta ( a k, b Nk ) distribution.

Another examples of a conjugate a priori distribution is present in a normal distribution with unknown expectation value when the a-priori distribution of a normal distribution with mean m and variance V ( where S is an arbitrary real number, and v is a positive real number). In this case, the posterior distribution is also a normal distribution.