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.
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.
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
Given as above, by itself follows an inverse gamma distribution:
while follows a t distribution with degrees of freedom. 
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:
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.
Normal distributions form an exponential family with natural parameters , , , and and sufficient statistics , , , and .
Measures difference between two distributions.
Posterior distribution of the parameters
See the articles on normal-gamma distribution and conjugate prior.