From Wikipedia, the free encyclopedia
Normal-WishartNotation |
|
---|
Parameters |
location (vector of
real) (real) scale matrix (
pos. def.) (real) |
---|
Support |
covariance matrix (
pos. def.) |
---|
PDF |
|
---|
In
probability theory and
statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous
probability distributions. It is the
conjugate prior of a
multivariate normal distribution with unknown
mean and
precision matrix (the inverse of the
covariance matrix).
[1]
Definition
Suppose
has a
multivariate normal distribution with
mean and
covariance matrix , where
has a
Wishart distribution. Then
has a normal-Wishart distribution, denoted as
Characterization
Probability density function
Properties
Scaling
Marginal distributions
By construction, the
marginal distribution over is a
Wishart distribution, and the
conditional distribution over given is a
multivariate normal distribution. The
marginal distribution over is a
multivariate t-distribution.
Posterior distribution of the parameters
After making observations , the posterior distribution of the parameters is
where
-
[2]
Generating normal-Wishart random variates
Generation of random variates is straightforward:
- Sample from a
Wishart distribution with parameters and
- Sample from a
multivariate normal distribution with mean and variance
Related distributions
Notes
References
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
|
---|
Discrete univariate | with finite support | |
---|
with infinite support | |
---|
|
---|
Continuous univariate | supported on a bounded interval | |
---|
supported on a semi-infinite interval | |
---|
supported on the whole real line | |
---|
with support whose type varies | |
---|
|
---|
Mixed univariate | |
---|
Multivariate (joint) | |
---|
Directional | |
---|
Degenerate and
singular | |
---|
Families | |
---|
|