In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-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 \mu and variance, where has an inverse-gamma distribution. Then has a normal-inverse-gamma distribution, denoted as (\text{NIG} 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 \mathbf{x} is a k \times 1 random vector, where is the determinant of the k \times k matrix \mathbf{V}. Note how this last equation reduces to the first form if k = 1 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 \mathbf{V} instead of its inverse as a parameter.

Cumulative distribution function

Properties

Marginal distributions

Given as above, \sigma^2 by itself follows an inverse gamma distribution: while follows a t distribution with 2 \alpha degrees of freedom. In the multivariate case, the marginal distribution of \mathbf{x} is a multivariate t distribution:

Summation

Scaling

Suppose Then for c>0, Proof: To prove this let and fix c>0. Defining, observe that the PDF of the random variable Y evaluated at (y_1,y_2) is given by 1/c^2 times the PDF of a random variable evaluated at. Hence the PDF of Y evaluated at (y_1,y_2) is given by : The right hand expression is the PDF for a random variable evaluated at (y_1,y_2), which completes the proof.

Exponential family

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

Information entropy

Kullback–Leibler divergence

Measures difference between two distributions.

Maximum likelihood estimation

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:

Related distributions