Probability density function
Probability density function of normal-inverse-gamma distribution for α = 1.0, 2.0 and 4.0, plotted in shifted and scaled coordinates.
Parameters location (real)

, for


, for
, for

, for

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.



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.

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. [1]

Proof for

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.

Exponential family

Normal 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

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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

Related distributions

See also


  1. ^ Murphy, Kevin P (2007). "Conjugate Bayesian analysis of the Gaussian distribution" (PDF). Retrieved 4 October 2021.((cite web)): CS1 maint: url-status (link)