In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions.

It is the conjugate prior of a normal distribution with unknown mean and precision.

[2] For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by meaning that the conditional distribution is a normal distribution with mean

Suppose also that the marginal distribution of T is given by where this means that T has a gamma distribution.

Here λ, α and β are parameters of the joint distribution.

Then (X,T) has a normal-gamma distribution, and this is denoted by The joint probability density function of (X,T) is where the conditional probability for

f ( x , τ ∣ μ , λ , α , β ) = f ( x ∣ τ , μ , λ , α , β ) f ( τ ∣ μ , λ , α , β )

By construction, the marginal distribution of

is a three-parameter non-standardized Student's t-distribution with parameters

) = ( 2 α , μ , β

[citation needed] The normal-gamma distribution is a four-parameter exponential family with natural parameters

[citation needed] The following moments can be easily computed using the moment generating function of the sufficient statistic:[3] where

( μ , λ , α , β ) ,

is distributed as[citation needed]

Assume that x is distributed according to a normal distribution with unknown mean

and that the prior distribution on

, has a normal-gamma distribution for which the density π satisfies Suppose i.e. the components of

The posterior distribution of

can be analytically determined by Bayes' theorem[4] explicitly, where

is the likelihood of the parameters given the data.

Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples: This expression can be simplified as follows: where

, the mean of the data samples, and

The posterior distribution of the parameters is proportional to the prior times the likelihood.

The final exponential term is simplified by completing the square.

On inserting this back into the expression above, This final expression is in exactly the same form as a Normal-Gamma distribution, i.e., The interpretation of parameters in terms of pseudo-observations is as follows: As a consequence, if one has a prior mean of

samples and a prior precision of

samples, the prior distribution over

, the posterior probability is Note that in some programming languages, such as Matlab, the gamma distribution is implemented with the inverse definition of

, so the fourth argument of the Normal-Gamma distribution is

Generation of random variates is straightforward: