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.

Suppose has a normal distribution with mean

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.

Note how this last equation reduces to the first form if

in which case the pdf becomes In the multivariate form, the corresponding change would be to regard the covariance matrix

( μ , λ , α , β )

by itself follows an inverse gamma distribution: while

probability density function is

∣ μ , α , β ) =

f ( x ∣ μ , α , β )

∣ μ , α , β )

Except for normalization factor, expression under the integral coincides with Inverse-gamma distribution

Substituting this expression and factoring dependence on

f ( x ∣ μ , α , β )

Shape of generalized Student's t-distribution is

In the multivariate case, the marginal distribution of

is a multivariate t distribution: Suppose Then for

( μ , λ , α , β )

, observe that the PDF of the random variable

( μ , λ , α , β )

random variable evaluated at

The right hand expression is the PDF for a

random variable evaluated at

Normal-inverse-gamma distributions form an exponential family with natural parameters

Measures difference between two distributions.

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

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

Generation of random variates is straightforward: