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 ...
, the inverse Wishart distribution, also called the inverted Wishart distribution, is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
defined on real-valued
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. In
Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
it is used as the
conjugate prior
In Bayesian probability theory, if the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions, and th ...
for the covariance matrix of a
multivariate normal
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
distribution.
We say
follows an inverse Wishart distribution, denoted as
, if its
inverse has a
Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions define ...
. Important identities have been derived for the inverse-Wishart distribution.
Density
The
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of the inverse Wishart is:
:
where
and
are
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
matrices,
is the determinant, and Γ
''p''(·) is the
multivariate gamma function.
Theorems
Distribution of the inverse of a Wishart-distributed matrix
If
and
is of size
, then
has an inverse Wishart distribution
.
Marginal and conditional distributions from an inverse Wishart-distributed matrix
Suppose
has an inverse Wishart distribution. Partition the matrices
and
conformably
Lithostratigraphy is a sub-discipline of stratigraphy, the geological science associated with the study of strata or rock layers. Major focuses include geochronology, comparative geology, and petrology.
In general, strata are primarily igne ...
with each other
:
where
and
are
matrices, then we have
#
is independent of
and
, where
is the
Schur complement In linear algebra and the theory of matrices, the Schur complement of a block matrix is defined as follows.
Suppose ''p'', ''q'' are nonnegative integers, and suppose ''A'', ''B'', ''C'', ''D'' are respectively ''p'' × ''p'', ''p'' × ''q'', ''q'' ...
of
in
;
#
;
#
, where
is a
matrix normal distribution
In statistics, the matrix normal distribution or matrix Gaussian distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.
Definition
The probability density ...
;
#
, where
;
Conjugate distribution
Suppose we wish to make inference about a covariance matrix
whose
prior
Prior (or prioress) is an ecclesiastical title for a superior in some religious orders. The word is derived from the Latin for "earlier" or "first". Its earlier generic usage referred to any monastic superior. In abbeys, a prior would be l ...
has a
distribution. If the observations