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 ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}/\lambda }$, where

has an inverse-gamma distribution. Then ${\displaystyle (x,\sigma ^{2})}$ has a normal-inverse-gamma distribution, denoted as

(${\displaystyle {\text{NIG}}}$ is also used instead of ${\displaystyle {\text{N-}}\Gamma ^{-1}.}$)

The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.

For the multivariate form where ${\displaystyle \mathbf {x} }$ is a ${\displaystyle k\times 1}$ random vector,

where ${\displaystyle |\mathbf {V} |}$ is the determinant of the ${\displaystyle k\times k}$ matrix ${\displaystyle \mathbf {V} }$. Note how this last equation reduces to the first form if ${\displaystyle k=1}$ so that ${\displaystyle \mathbf {x} ,\mathbf {V} ,{\boldsymbol {\mu }}}$ are scalars.