A priori distribution

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The a priori distribution is a term from Bayesian statistics .

definition

The situation is as follows: is an unknown population parameter that is to be estimated on the basis of observations of a random variable . ${\ displaystyle \ theta}$ ${\ displaystyle x}$ ${\ displaystyle X}$

Given a distribution for the parameter that describes the knowledge of the parameter before observing the sample. This distribution is called the a priori distribution. ${\ displaystyle \ theta}$ ${\ displaystyle \ theta}$

Furthermore, let the conditional distribution of the sample be given under the condition , which is also known as the likelihood function . ${\ displaystyle \ theta = \ theta _ {0}}$

From the a priori distribution and the likelihood function, the a posteriori distribution can be calculated with the help of Bayes' theorem , which is fundamental for the calculation of point estimates (see Bayesian estimates ) and interval estimates in Bayesian statistics (see credibility interval ) is.

(Non-) informative a priori distributions

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

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

The concept of non-informative a priori distribution is explained using an example: Let the random variable Y be the mean intelligence quotient in the city of ZZZ. Due to the construction of the intelligence quotient it is known that Y is normally distributed with standard deviation 15 and unknown parameter . The intelligence quotient is measured on a sample of N volunteers. An arithmetic mean of 105 is observed in this sample . ${\ displaystyle \ mu}$

A non-informative a priori distribution is given in this case by

{\ displaystyle p (\ theta) \ propto c}

,

where is a positive real number. In this way, the a posteriori distribution is a normal distribution with a mean value of 105 and standard deviation . 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. ${\ displaystyle c <\ infty}$ ${\ displaystyle {\ frac {15} {\ sqrt {N}}}}$ ${\ displaystyle \ mu}$

Actual vs. improper a priori distributions

The above example illustrates a problem that often occurs when using non-informative a priori distributions: defines a so-called improper a priori distribution. Improper a priori distributions are characterized by the fact that the integral of the a priori distribution is greater than 1. Hence improper a priori distributions are not probability distributions. In many cases, however, it can be shown that the a posteriori distribution is defined even when using an improper distribution. This is true when ${\ displaystyle p (\ theta) \ propto c}$

{\ displaystyle \ int p (\ theta | y) \ mathrm {d} \ theta <\ infty}

applies to all . An actual a priori distribution is defined by the fact that it is independent of the data and that its integral is 1. ${\ displaystyle y}$ ${\ displaystyle p (\ theta)}$

Conjugated a priori distributions

A priori and a posteriori distributions are conjugate for a given likelihood function if they have the same distribution type.

An example of this is the binomial beta model: be a binomial random variable with a probability of success as a parameter. Successes are observed in individual experiments . As a priori distribution for one is distribution to use. Under these conditions, the a posteriori distribution is a distribution. ${\ displaystyle X}$ ${\ displaystyle p \ in (0,1)}$ ${\ displaystyle N}$ ${\ displaystyle k}$ ${\ displaystyle p}$ ${\ displaystyle \ mathrm {B} (\ alpha, \ beta)}$ ${\ displaystyle (0,1)}$ ${\ displaystyle \ mathrm {B} (\ alpha + k, \ beta + Nk)}$

Another example is the update of a normally distributed prior with a Gaussian likelihood function. The a posteriori distribution is then also a normal distribution.

literature

James O. Berger: Statistical decision theory and Bayesian analysis . Springer Series in Statistics, Springer-Verlag, New York Berlin Heidelberg 1985. ISBN 0-387-96098-8
Andrew Gelman et al .: Bayesian Data Analysis . Chapman & Hall / CRC, Boca Raton London New York Washington DC 2013. ISBN 978-1439840955