In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, and specifically in the study of the
Dirichlet distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymb ...
, a neutral vector of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s is one that exhibits a particular type of
statistical independence
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of ...
amongst its elements.
[ ] In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the element that was omitted.
Definition
A single element
of a random vector
is neutral if the ''relative'' proportions of all the other elements are independent of
.
Formally, consider the vector of random variables
:
where
:
The values
are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say
, and consider the distribution of the remaining intervals within the remaining length. The first element of
, viz
is defined as ''neutral'' if
is
statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
of the vector
:
Variable
is neutral if
is independent of the remaining interval: that is,
being independent of
:
Thus
, viewed as the first element of
, is neutral.
In general, variable
is neutral if
is independent of
:
Complete neutrality
A vector for which each element is neutral is completely neutral.
If
is drawn from a Dirichlet distribution, then
is completely neutral. In 1980, James and Mosimann
showed that the Dirichlet distribution is characterised by neutrality.
See also
*
Generalized Dirichlet distribution In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely ...
References
{{reflist
Theory of probability distributions
Independence (probability theory)