In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
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 normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous
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 ...
s. It is 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 ...
of a
multivariate normal distribution with unknown
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
and
precision matrix (the inverse of the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
).
[Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media. Page 690.]
Definition
Suppose
:
has a
multivariate normal distribution with
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
and
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
, where
:
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 ...
. Then
has a normal-Wishart distribution, denoted as
:
Characterization
Probability density function
:
Properties
Scaling
Marginal distributions
By construction, the
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
over
is 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 ...
, and the
conditional distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the co ...
over
given
is a
multivariate normal distribution. The
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
over
is a
multivariate ''t''-distribution.
Posterior distribution of the parameters
After making
observations
, the posterior distribution of the parameters is
:
where
:
:
:
:
[Cross Validated, https://stats.stackexchange.com/q/324925]
Generating normal-Wishart random variates
Generation of random variates is straightforward:
# Sample
from 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 ...
with parameters
and
# Sample
from a
multivariate normal distribution with mean
and variance
Related distributions
* The
normal-inverse Wishart distribution
In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate n ...
is essentially the same distribution parameterized by variance rather than precision.
* The
normal-gamma distribution
In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean ...
is the one-dimensional equivalent.
* The
multivariate normal distribution and
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 ...
are the component distributions out of which this distribution is made.
Notes
References
* Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media.
{{DEFAULTSORT:Normal-Wishart Distribution
Multivariate continuous distributions
Conjugate prior distributions
Normal distribution