Normal-inverse-gamma distribution

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.

Probability density function
Parameters location (real)

, for


, for
, for

, for



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 )

In a multivariate form of the normal-inverse-gamma distribution, – that is, conditional on , is a random vector that follows the multivariate normal distribution with mean and covariance – while, as in the univariate case, .


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


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:



Exponential family

Information entropy

Kullback–Leibler divergence

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:

  1. Sample from an inverse gamma distribution with parameters and
  2. Sample from a normal distribution with mean and variance
  • The normal-gamma distribution is the same distribution parameterized by precision rather than variance
  • A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution


  • Denison, David G. T. ; Holmes, Christopher C.; Mallick, Bani K.; Smith, Adrian F. M. (2002) Bayesian Methods for Nonlinear Classification and Regression, Wiley. ISBN 0471490369
  • Koch, Karl-Rudolf (2007) Introduction to Bayesian Statistics (2nd Edition), Springer. ISBN 354072723X
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