Definition
Suppose

has a normal distribution with mean
and variance
, where

has an inverse gamma distribution. Then
has a normal-inverse-gamma distribution, denoted as

(
is also used instead of
)
The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.
Characterization
Probability density function

For the multivariate form where
is a
random vector,

where
is the determinant of the
matrix
. Note how this last equation reduces to the first form if
so that
are scalars.
Alternative parameterization
It is also possible to let
in which case the pdf becomes

In the multivariate form, the corresponding change would be to regard the covariance matrix
instead of its inverse
as a parameter.
Cumulative distribution function

Properties
Marginal distributions
Given
as above,
by itself follows an inverse gamma distribution:

while
follows a t distribution with
degrees of freedom. [1]
Proof for
For
probability density function is
Marginal distribution over
is
Except for normalization factor, expression under the integral coincides with Inverse-gamma distribution
with
,
,
.
Since
, and
Substituting this expression and factoring dependence on
,
Shape of generalized Student's t-distribution is
.
Marginal distribution
follows t-distribution with
degrees of freedom
.
In the multivariate case, the marginal distribution of
is a multivariate t distribution:

Summation
Scaling
Suppose

Then for
,

Proof: To prove this let
and fix
. Defining
, observe that the PDF of the random variable
evaluated at
is given by
times the PDF of a
random variable evaluated at
. Hence the PDF of
evaluated at
is given by :
The right hand expression is the PDF for a
random variable evaluated at
, which completes the proof.
Exponential family
Normal distributions form an exponential family with natural parameters
,
,
, and
and sufficient statistics
,
,
, and
.
Information entropy
Kullback–Leibler divergence
Measures difference between two distributions.
Posterior distribution of the parameters
See the articles on normal-gamma distribution and conjugate prior.