# Scaled inverse chi-squared distribution

The scaled inverse chi-squared distribution $\psi \,{\mbox{inv-}}\chi ^{2}(\nu )$, where $\psi$ is the scale parameter, equals the univariate inverse Wishart distribution ${\mathcal {W}}^{-1}(\psi ,\nu )$ with degrees of freedom $\nu$.
This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution:
If $X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )$ then $X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )$ as well as $\psi /X\sim \chi ^{2}(\nu )$ and $1/X\sim \psi ^{-1}\chi ^{2}(\nu )$.
Instead of $\psi$, the scaled inverse chi-squared distribution is however most frequently parametrized by the scale parameter $\tau ^{2}=\psi /\nu$ and the distribution $\nu \tau ^{2}\,{\mbox{inv-}}\chi ^{2}(\nu )$ is denoted by ${\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$.
In terms of $\tau ^{2}$ the above relations can be written as follows:
If $X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$ then ${\frac {X}{\nu \tau ^{2}}}\sim {\mbox{inv-}}\chi ^{2}(\nu )$ as well as ${\frac {\nu \tau ^{2}}{X}}\sim \chi ^{2}(\nu )$ and $1/X\sim {\frac {1}{\nu \tau ^{2}}}\chi ^{2}(\nu )$.
This family of scaled inverse chi-squared distributions is a reparametrization of the inverse-gamma distribution.
Specifically, if
    $X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )={\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$ then $X\sim {\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\psi }{2}}\right)={\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\nu \tau ^{2}}{2}}\right)$
Either form may be used to represent the maximum entropy distribution for a fixed first inverse moment $(E(1/X))$ and first logarithmic moment $(E(\ln(X))$.
The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics. Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution. The same prior in alternative parametrization is given by the inverse-gamma distribution.

## Characterization

The probability density function of the scaled inverse chi-squared distribution extends over the domain $x>0$ and is
    $f(x;\nu ,\tau ^{2})={\frac {(\tau ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \tau ^{2}}{2x}}\right]}{x^{1+\nu /2}}}$
where $\nu$ is the degrees of freedom parameter and $\tau ^{2}$ is the scale parameter. The cumulative distribution function is
    $F(x;\nu ,\tau ^{2})=\Gamma \left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)\left/\Gamma \left({\frac {\nu }{2}}\right)\right.$
    $=Q\left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)$
where $\Gamma (a,x)$ is the incomplete gamma function, $\Gamma (x)$ is the gamma function and $Q(a,x)$ is a regularized gamma function. The characteristic function is
    $\varphi (t;\nu ,\tau ^{2})=$
    ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\tau ^{2}\nu t}}\right),$
where $K_{\frac {\nu }{2}}(z)$ is the modified Modified Bessel function of the second kind.

## Parameter estimation

The maximum likelihood estimate of $\tau ^{2}$ is
    $\tau ^{2}=n/\sum _{i=1}^{n}{\frac {1}{x_{i}}}.$
The maximum likelihood estimate of ${\frac {\nu }{2}}$ can be found using Newton's method on:
    $\ln \left({\frac {\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)={\frac {1}{n}}\sum _{i=1}^{n}\ln \left(x_{i}\right)-\ln \left(\tau ^{2}\right),$
where $\psi (x)$ is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for $\nu .$ Let ${\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$ be the sample mean. Then an initial estimate for $\nu$ is given by:
    ${\frac {\nu }{2}}={\frac {\bar {x}}{{\bar {x}}-\tau ^{2}}}.$

## Bayesian estimation of the variance of a normal distribution

The scaled inverse chi-squared distribution has a second important application, in the Bayesian estimation of the variance of a Normal distribution.
According to Bayes' theorem, the posterior probability distribution for quantities of interest is proportional to the product of a prior distribution for the quantities and a likelihood function:
    $p(\sigma ^{2}|D,I)\propto p(\sigma ^{2}|I)\;p(D|\sigma ^{2})$
where D represents the data and I represents any initial information about σ$^2$ that we may already have.
The simplest scenario arises if the mean μ is already known; or, alternatively, if it is the conditional distribution of σ$^2$ that is sought, for a particular assumed value of μ.
Then the likelihood term L(σ$^2$|D) = p(D|σ$^2$) has the familiar form
    ${\mathcal {L}}(\sigma ^{2}|D,\mu )={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{n}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]$
Combining this with the rescaling-invariant prior p(σ$^2$|I) = 1/σ$^2$, which can be argued (e.g. following Jeffreys) to be the least informative possible prior for σ$^2$ in this problem, gives a combined posterior probability
    $p(\sigma ^{2}|D,I,\mu )\propto {\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]$
This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ$^2$ = s$^2$ = (1/n) Σ (x$_i$-μ)$^2$
Gelman and co-authors remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior "this result is not surprising."
In particular, the choice of a rescaling-invariant prior for σ$^2$ has the result that the probability for the ratio of σ$^2$ / s$^2$ has the same form (independent of the conditioning variable) when conditioned on s$^2$ as when conditioned on σ$^2$:
    $p({\tfrac {\sigma ^{2}}{s^{2}}}|s^{2})=p({\tfrac {\sigma ^{2}}{s^{2}}}|\sigma ^{2})$
In the sampling-theory case, conditioned on σ$^2$, the probability distribution for (1/s$^2$) is a scaled inverse chi-squared distribution; and so the probability distribution for σ$^2$ conditioned on s$^2$, given a scale-agnostic prior, is also a scaled inverse chi-squared distribution.

### Use as an informative prior

If more is known about the possible values of σ$^2$, a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ$^2$(n$_0$, s$_0$$^2$) can be a convenient form to represent a more informative prior for σ$^2$, as if from the result of n$_0$ previous observations (though n$_0$ need not necessarily be a whole number):
    $p(\sigma ^{2}|I^{\prime },\mu )\propto {\frac {1}{\sigma ^{n_{0}+2}}}\;\exp \left[-{\frac {n_{0}s_{0}^{2}}{2\sigma ^{2}}}\right]$
Such a prior would lead to the posterior distribution
    $p(\sigma ^{2}|D,I^{\prime },\mu )\propto {\frac {1}{\sigma ^{n+n_{0}+2}}}\;\exp \left[-{\frac {ns^{2}+n_{0}s_{0}^{2}}{2\sigma ^{2}}}\right]$
which is itself a scaled inverse chi-squared distribution. The scaled inverse chi-squared distributions are thus a convenient conjugate prior family for σ$^2$ estimation.

### Estimation of variance when mean is unknown

If the mean is not known, the most uninformative prior that can be taken for it is arguably the translation-invariant prior p(μ|I) ∝ const., which gives the following joint posterior distribution for μ and σ$^2$,
    ${\begin{aligned}p(\mu ,\sigma ^{2}\mid D,I)&\propto {\frac {1}{\sigma ^{n+2}}}\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{\sigma ^{n+2}}}\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\exp \left[-{\frac {n(\mu -{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\end{aligned}}$
The marginal posterior distribution for σ$^2$ is obtained from the joint posterior distribution by integrating out over μ,
    ${\begin{aligned}p(\sigma ^{2}|D,I)\;\propto \;&{\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\;\int _{-\infty }^{\infty }\exp \left[-{\frac {n(\mu -{\bar {x}})^{2}}{2\sigma ^{2}}}\right]d\mu \\=\;&{\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\;{\sqrt {2\pi \sigma ^{2}/n}}\\\propto \;&(\sigma ^{2})^{-(n+1)/2}\;\exp \left[-{\frac {(n-1)s^{2}}{2\sigma ^{2}}}\right]\end{aligned}}$
This is again a scaled inverse chi-squared distribution, with parameters $\scriptstyle {n-1}\;$ and $\scriptstyle {s^{2}=\sum (x_{i}-{\bar {x}})^{2}/(n-1)}$.

## Related distributions

* If $X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$ then $kX\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,k\tau ^{2})\,$
* If $X\sim {\mbox{inv-}}\chi ^{2}(\nu )\,$ (Inverse-chi-squared distribution) then $X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,1/\nu )\,$
* If $X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$ then ${\frac {X}{\tau ^{2}\nu }}\sim {\mbox{inv-}}\chi ^{2}(\nu )\,$ (Inverse-chi-squared distribution)
* If $X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})$ then $X\sim {\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\nu \tau ^{2}}{2}}\right)$ (Inverse-gamma distribution)
* Scaled inverse chi square distribution is a special case of type 5 Pearson distribution