Normal-inverse-gamma distribution


In probability theory and statistics, the normal-inverse-gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

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
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.

Properties

Marginal distributions

Given as above, by itself follows an inverse gamma distribution:
while follows a t distribution with degrees of freedom.
In the multivariate case, the marginal distribution of is a multivariate t distribution:

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-inverse-gamma distributions form an exponential family with natural parameters,,, and and sufficient statistics,,, and.

Kullback–Leibler divergence

Measures difference between two distributions.

Posterior distribution of the parameters

See the articles on normal-gamma distribution and conjugate prior.

Interpretation of the parameters

See the articles on normal-gamma distribution and conjugate prior.

Generating normal-inverse-gamma random variates

Generation of random variates is straightforward:
  1. Sample from an inverse gamma distribution with parameters and
  2. Sample from a normal distribution with mean and variance

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