Dirichlet distribution

The Dirichletverteilung (after Peter Gustav Lejeune Dirichlet ) is a family of continuous multivariate probability distributions.

It is the multivariate extension of the beta distribution and the conjugate a priori distribution of the multinomial distribution in Bayesian statistics. Its density function indicates the probability of K different, exclusive events, with each event is sometimes observed.

Illustration

The multinomial distribution gives the probabilities for k to different events, eg how likely it is, a One, Two, Three, Four, to roll the dice in a throw five or six. In contrast, specifies the Dirichlet distribution, how likely such a distribution occurs. In the case of a cube factory the Dirichlet distribution could indicate so, how likely are the distributions of the cube results in the fabricated dice. Functioning of the machines of the cube factory correct, the probability of anything but the uniform distribution would be (all eyes numbers are equally likely ) is very low. This would correspond to a parameter vector with the same and very high elements such as. Contrast, would mean that the machine dies fabricate, in which the eyes number one twice as frequently as any other face value. And this is because the values ​​in turn are almost all very high and thus the variance low. Would, for example, the values ​​in but all, then dice would be produced which have a strong tendency to face value. What is the preferred A dice, would be by chance, as all values ​​are equal. The smaller the value, the greater would be the unfairness of most dice, and the less likely would die without a preferred face value.

Density function

The Dirichletverteilung of order K ≥ 2 with parameters has the following density function:

For all with and. Therefore, the sum of all probabilities is equal to 1

The normalizing constant is the multinomial beta function, which can be represented by gamma functions: