The scaled inverse chi-squared distribution ψ inv- χ 2 ( ν ) {\displaystyle \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} , where ψ {\displaystyle \psi } is the scale parameter, equals the univariate inverse Wishart distribution
W − 1 ( ψ , ν ) {\displaystyle {\mathcal {W}}^{-1}(\psi ,\nu )} with degrees of freedom ν {\displaystyle \nu } . This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution: If X ∼ ψ inv- χ 2 ( ν ) {\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )} then X / ψ ∼ inv- χ 2 ( ν ) {\displaystyle X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )} as well as ψ / X ∼ χ 2 ( ν ) {\displaystyle \psi /X\sim \chi ^{2}(\nu )} and 1 / X ∼ ψ − 1 χ 2 ( ν ) {\displaystyle 1/X\sim \psi ^{-1}\chi ^{2}(\nu )} . Instead of ψ {\displaystyle \psi } , the scaled inverse chi-squared distribution is however most frequently parametrized by the scale parameter τ 2 = ψ / ν {\displaystyle \tau ^{2}=\psi /\nu } and the distribution ν τ 2 inv- χ 2 ( ν ) {\displaystyle \nu \tau ^{2}\,{\mbox{inv-}}\chi ^{2}(\nu )} is denoted by Scale-inv- χ 2 ( ν , τ 2 ) {\displaystyle {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})} .
In terms of τ 2 {\displaystyle \tau ^{2}} the above relations can be written as follows: If X ∼ Scale-inv- χ 2 ( ν , τ 2 ) {\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})} then X ν τ 2 ∼ inv- χ 2 ( ν ) {\displaystyle {\frac {X}{\nu \tau ^{2}}}\sim {\mbox{inv-}}\chi ^{2}(\nu )} as well as ν τ 2 X ∼ χ 2 ( ν ) {\displaystyle {\frac {\nu \tau ^{2}}{X}}\sim \chi ^{2}(\nu )} and 1 / X ∼ 1 ν τ 2 χ 2 ( ν ) {\displaystyle 1/X\sim {\frac {1}{\nu \tau ^{2}}}\chi ^{2}(\nu )} .
This family of scaled inverse chi-squared distributions is a reparametrization of the inverse-gamma distribution. Specifically, if
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