Scaled inverse chi-squared distribution

Scaled inverse chi-squared
Probability density function
Cumulative distribution function
Parameters	$\nu >0$ ${\tau }^2>0$
Support	$x\in (0,\mathrm{\infty })$
PDF	$\frac{({\tau }^2\nu /2{)}^{\nu /2}}{\Gamma (\nu /2)}\frac{\exp \left[\frac{-\nu {\tau }^2}{2x}\right]}{x^{1+\nu /2}}$
CDF	$\Gamma (\frac{\nu }{2},\frac{{\tau }^2\nu }{2x})/\Gamma \left(\frac{\nu }{2}\right)$
Mean	$\frac{\nu {\tau }^2}{\nu -2}$ for $\nu >2$
Mode	$\frac{\nu {\tau }^2}{\nu +2}$
Variance	$\frac{2{\nu }^2{\tau }^4}{(\nu -2{)}^2(\nu -4)}$for $\nu >4$
Skewness	$\frac{4}{\nu -6}\sqrt{2(\nu -4)}$for $\nu >6$
Excess kurtosis	$\frac{12(5\nu -22)}{(\nu -6)(\nu -8)}$for $\nu >8$
Entropy	$\frac{\nu }{2}+\ln \left(\frac{{\tau }^2\nu }{2}\Gamma \left(\frac{\nu }{2}\right)\right)$ $-(1+\frac{\nu }{2})\psi \left(\frac{\nu }{2}\right)$
MGF	$\frac{2}{\Gamma (\frac{\nu }{2})}{\left(\frac{-{\tau }^2\nu t}{2}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2{\tau }^2\nu t}\right)$
CF	$\frac{2}{\Gamma (\frac{\nu }{2})}{\left(\frac{-i{\tau }^2\nu t}{2}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2i{\tau }^2\nu t}\right)$

The scaled inverse chi-squared distribution $\psi \text{inv-}{\chi }^2(\nu )$, where $\psi$ is the scale parameter, equals the univariate inverse Wishart distribution ${𝒲}^{-1}(\psi ,\nu )$ 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 $X\sim \psi \text{inv-}{\chi }^2(\nu )$ then $X/\psi \sim \text{inv-}{\chi }^2(\nu )$ 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 $\nu {\tau }^2\text{inv-}{\chi }^2(\nu )$ is denoted by $\text{Scale-inv-}{\chi }^2(\nu ,{\tau }^2)$.

In terms of ${\tau }^2$ the above relations can be written as follows:

If $X\sim \text{Scale-inv-}{\chi }^2(\nu ,{\tau }^2)$ then $\frac{X}{\nu {\tau }^2}\sim \text{inv-}{\chi }^2(\nu )$ 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

$X\sim \psi \text{inv-}{\chi }^2(\nu )=\text{Scale-inv-}{\chi }^2(\nu ,{\tau }^2)$   then   $X\sim \text{Inv-Gamma}(\frac{\nu }{2},\frac{\psi }{2})=\text{Inv-Gamma}(\frac{\nu }{2},\frac{\nu {\tau }^2}{2})$

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

$F(x;\nu ,{\tau }^2)=\Gamma (\frac{\nu }{2},\frac{{\tau }^2\nu }{2x})/\Gamma \left(\frac{\nu }{2}\right)$

$=Q(\frac{\nu }{2},\frac{{\tau }^2\nu }{2x})$

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

$\phi (t;\nu ,{\tau }^2)=$

$\frac{2}{\Gamma (\frac{\nu }{2})}{\left(\frac{-i{\tau }^2\nu t}{2}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2i{\tau }^2\nu t}\right),$

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 $\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$ be the sample mean. Then an initial estimate for $\nu$ is given by:

$\frac{\nu }{2}=\frac{\overline{x}}{\overline{x}-{\tau }^2}.$

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 σ² 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 σ² that is sought, for a particular assumed value of μ.

Then the likelihood term L(σ²|D) = p(D|σ²) has the familiar form

$ℒ({\sigma }^2|D,\mu )=\frac{1}{{\left(\sqrt{2\pi }\sigma \right)}^n}\exp [-\frac{\sum _i^n(x_i-\mu {)}^2}{2{\sigma }^2}]$

Combining this with the rescaling-invariant prior p(σ²|I) = 1/σ², which can be argued (e.g. following Jeffreys) to be the least informative possible prior for σ² in this problem, gives a combined posterior probability

$p({\sigma }^2|D,I,\mu )\propto \frac{1}{{\sigma }^{n+2}}\exp [-\frac{\sum _i^n(x_i-\mu {)}^2}{2{\sigma }^2}]$

This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ² = s² = (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 σ² has the result that the probability for the ratio of σ² / s² has the same form (independent of the conditioning variable) when conditioned on s² as when conditioned on σ²:

$p(\frac{{\sigma }^2}{s^2}|s^2)=p(\frac{{\sigma }^2}{s^2}|{\sigma }^2)$

In the sampling-theory case, conditioned on σ², the probability distribution for (1/s²) is a scaled inverse chi-squared distribution; and so the probability distribution for σ² conditioned on s², 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 σ², a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ²(n₀, s₀²) can be a convenient form to represent a more informative prior for σ², as if from the result of n₀ previous observations (though n₀ need not necessarily be a whole number):

$p({\sigma }^2|I^{\prime },\mu )\propto \frac{1}{{\sigma }^{n_0+2}}\exp [-\frac{n_0{s}_0^2}{2{\sigma }^2}]$

Such a prior would lead to the posterior distribution

$p({\sigma }^2|D,I^{\prime },\mu )\propto \frac{1}{{\sigma }^{n+n_0+2}}\exp [-\frac{n{s}^2+n_0{s}_0^2}{2{\sigma }^2}]$

which is itself a scaled inverse chi-squared distribution. The scaled inverse chi-squared distributions are thus a convenient conjugate prior family for σ² 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 σ²,

$\begin{array}{cc}p(\mu ,{\sigma }^2\mid D,I)& \propto \frac{1}{{\sigma }^{n+2}}\exp [-\frac{\sum _i^n(x_i-\mu {)}^2}{2{\sigma }^2}]\\ & =\frac{1}{{\sigma }^{n+2}}\exp [-\frac{\sum _i^n(x_i-\overline{x}{)}^2}{2{\sigma }^2}]\exp [-\frac{n(\mu -\overline{x}{)}^2}{2{\sigma }^2}]\end{array}$

The marginal posterior distribution for σ² is obtained from the joint posterior distribution by integrating out over μ,

$\begin{array}{cc}p({\sigma }^2|D,I)\propto & \frac{1}{{\sigma }^{n+2}}\exp [-\frac{\sum _i^n(x_i-\overline{x}{)}^2}{2{\sigma }^2}]\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\exp [-\frac{n(\mu -\overline{x}{)}^2}{2{\sigma }^2}]d\mu \\ =& \frac{1}{{\sigma }^{n+2}}\exp [-\frac{\sum _i^n(x_i-\overline{x}{)}^2}{2{\sigma }^2}]\sqrt{2\pi {\sigma }^2/n}\\ \propto & ({\sigma }^2{)}^{-(n+1)/2}\exp [-\frac{(n-1)s^2}{2{\sigma }^2}]\end{array}$

This is again a scaled inverse chi-squared distribution, with parameters $n-1$ and $s^2=\sum (x_i-\overline{x}{)}^2/(n-1)$.

Related distributions

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