Scaled inverse chi-squared distribution

Scaled inverse chi-squared

Probability density function
Cumulative distribution function
Parameters	$\nu > 0\,$ $\tau^2 > 0\,$
Support	$x \in (0, \infty)$
PDF	$\frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \tau^2}{2 x}\right]}{x^{1+\nu/2}}$
CDF	$\Gamma\left(\frac{\nu}{2},\frac{\tau^2\nu}{2x}\right) \left/\Gamma\left(\frac{\nu}{2}\right)\right.$
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 $\nu >6\,$
Ex. kurtosis	$\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}$for $\nu >8\,$
Entropy	$\frac{\nu}{2} \!+\!\ln\left(\frac{\tau^2\nu}{2}\Gamma\left(\frac{\nu}{2}\right)\right)$ $\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\left(\frac{\nu}{2}\right)$
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 is the distribution for x = 1/s², where s² is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ² = τ². The distribution is therefore parametrised by the two quantities ν and τ², referred to as the number of chi-squared degrees of freedom and the scaling parameter, respectively.

This family of scaled inverse chi-squared distributions is closely related to two other distribution families, those of the inverse-chi-squared distribution and the inverse gamma distribution. Compared to the inverse-chi-squared distribution, the scaled distribution has an extra parameter τ², which scales the distribution horizontally and vertically, representing the inverse-variance of the original underlying process. Also, the scale inverse chi-squared distribution is presented as the distribution for the inverse of the mean of ν squared deviates, rather than the inverse of their sum. The two distributions thus have the relation that if

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

Compared to the inverse gamma distribution, the scaled inverse chi-squared distribution describes the same data distribution, but using a different parametrization, which may be more convenient in some circumstances. Specifically, if

$X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2)$   then   $X \sim \textrm{Inv-Gamma}\left(\frac{\nu}{2}, \frac{\nu\tau^2}{2}\right)$

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, somewhat unrelated to its use as a predictive distribution for x = 1/s². Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution. In this context the scaling parameter is denoted by σ₀² rather than by τ², and has a different interpretation. The application has been more usually presented using the inverse gamma distribution formulation instead; however, some authors, following in particular Gelman et al. (1995/2004) argue that the inverse chi-squared parametrisation is more intuitive.

Characterization

The probability density function of the scaled inverse chi-squared distribution extends over the domain $x>0$ and is

$f(x; \nu, \tau^2)= \frac{(\tau^2\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~ \frac{\exp\left[ \frac{-\nu \tau^2}{2 x}\right]}{x^{1+\nu/2}}$

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 Bessel function of the second kind.

Differential equation

$\left\{2 x^2 f'(x)+f(x) \left(-\nu \tau ^2+\nu x+2 x\right)=0,f(1)=\frac{2^{-\nu /2} e^{-\frac{\nu \tau ^2}{2}} \left(\nu \tau ^2\right)^{\nu /2}}{\Gamma \left(\frac{\nu }{2}\right)}\right\}$

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:

$\ln(\frac{\nu}{2}) + \psi(\frac{\nu}{2}) = \sum_{i=1}^n \ln(x_i) - n \ln(\tau^2) ,$

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 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

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

Combining this with the rescaling-invariant prior p(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

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

This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ² = s² = (1/n) Σ (x_i-μ)²

Gelman et al remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior the "result is not surprising".^[1]

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:

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

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 less uninformative prior for σ², as if from the result of n₀ previous observations (though n₀ need not necessarily be a whole number):

$p(\sigma^2|I^\prime, \mu) \propto \frac{1}{\sigma^{n_0+2}} \; \exp \left[ -\frac{n_0 s_0^2}{2\sigma^2} \right]$

Such a prior would lead to the posterior distribution

$p(\sigma^2|D, I^\prime, \mu) \propto \frac{1}{\sigma^{n+n_0+2}} \; \exp \left[ -\frac{\sum{ns^2 + n_0 s_0^2}}{2\sigma^2} \right]$

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

$\begin{align} p(\mu, \sigma^2 \mid D, I) & \propto \frac{1}{\sigma^{n+2}} \exp \left[ -\frac{\sum_i^n(x_i-\mu)^2}{2\sigma^2} \right] \\ & = \frac{1}{\sigma^{n+2}} \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \exp \left[ -\frac{\sum_i^n(\mu -\bar{x})^2}{2\sigma^2} \right] \end{align}$

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

$\begin{align} p(\sigma^2|D, I) \; \propto \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \; \int_{-\infty}^{\infty} \exp \left[ -\frac{\sum_i^n(\mu -\bar{x})^2}{2\sigma^2} \right] d\mu\\ = \; & \frac{1}{\sigma^{n+2}} \; \exp \left[ -\frac{\sum_i^n(x_i-\bar{x})^2}{2\sigma^2} \right] \; \sqrt{2 \pi \sigma^2 / n} \\ \propto \; & (\sigma^2)^{-(n+1)/2} \; \exp \left[ -\frac{(n-1)s^2}{2\sigma^2} \right] \end{align}$

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 $k X \sim \mbox{Scale-inv-}\chi^2(\nu, k \tau^2)\,$
• 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

References

• Gelman A. et al (1995), Bayesian Data Analysis, pp 474–475; also pp 47, 480

1. Gelman et al (1995), Bayesian Data Analysis (1st ed), p.68