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.
Definition Suppose
x ∣ σ 2 , μ , λ ∼ N ( μ , σ 2 / λ ) {\displaystyle x\mid \sigma ^{2},\mu ,\lambda \sim \mathrm {N} (\mu ,\sigma ^{2}/\lambda )\,\!}
has a normal distribution with mean μ {\displaystyle \mu } and variance σ 2 / λ {\displaystyle \sigma ^{2}/\lambda } , where
σ 2 ∣ α , β ∼ Γ − 1 ( α , β ) {\displaystyle \sigma ^{2}\mid \alpha ,\beta \sim \Gamma ^{-1}(\alpha ,\beta )\!}
has an inverse-gamma distribution. Then ( x , σ 2 ) {\displaystyle (x,\sigma ^{2})} has a normal-inverse-gamma distribution, denoted as
( x , σ 2 ) ∼ N- Γ − 1 ( μ , λ , α , β ) . {\displaystyle (x,\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )\!.}
( NIG {\displaystyle {\text{NIG}}} is also used instead of N- Γ − 1 . {\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.
Characterization
Probability density function
f ( x , σ 2 ∣ μ , λ , α , β ) = λ σ 2 π β α Γ ( α ) ( 1 σ 2 ) α + 1 exp ( − 2 β + λ ( x − μ ) 2 2 σ 2 ) {\displaystyle f(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {\sqrt {\lambda }}{\sigma {\sqrt {2\pi }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)}
For the multivariate form where x {\displaystyle \mathbf {x} } is a k × 1 {\displaystyle k\times 1} random vector,
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