Scaled-inverse-chi-squared Distribution

The scaled inverse chi-squared distribution is the distribution for ''x'' = 1/''s''², where ''s''² 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 ''τ''², which scales the distribution horizontally and vertically, representing the inverse-variance of the original underlying process. Also, the scaled 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
:$X \sim \mbox\chi^2(\nu, \tau^2)$   then   $\frac \sim \mbox\chi^2(\nu)$
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
:$X \sim \mbox\chi^2(\nu, \tau^2)$   then   $X \sim \textrm\left(\frac, \frac\right)$
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/''s''². 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

:$f(x; \nu, \tau^2)= \frac~ \frac$

where $\nu$ is the degrees of freedom parameter and $\tau^2$ is the scale parameter. The cumulative distribution function is

:$F(x; \nu, \tau^2)= \Gamma\left(\frac,\frac\right) \left/\Gamma\left(\frac\right)\right.$
:$=Q\left(\frac,\frac\right)$

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

:$\varphi(t;\nu,\tau^2)=$
:$\frac\left(\frac\right)^\!\!K_\left(\sqrt\right) ,$

where $K_(z)$ is the modified Bessel function of the second kind.

Parameter estimation

The maximum likelihood estimate of $\tau^2$ is

:$\tau^2 = n/\sum_^n \frac.$

The maximum likelihood estimate of $\frac$ can be found using Newton's method on:

:$\ln\left(\frac\right) - \psi\left(\frac\right) = \frac \sum_^n \ln\left(x_i\right) - \ln\left(\tau^2\right) ,$

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 $\bar = \frac\sum_^n x_i$ be the sample mean. Then an initial estimate for $\nu$ is given by:

:$\frac = \frac.$

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(\sigma^2, D,\mu) = \frac \; \exp \left -\frac \right/math>

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 \; \exp \left -\frac \right/math>
This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = ''n'' and τ² = ''s''² = (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".Gelman ''et al'' (1995), ''Bayesian Data Analysis'' (1st ed), p.68

In particular, the choice of a rescaling-invariant prior for σ² has the result that the probability for the ratio of σ² / ''s''² has the same form (independent of the conditioning variable) when conditioned on ''s''² as when conditioned on σ²:

:$p(\tfrac, s^2) = p(\tfrac, \sigma^2)$

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 ''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 σ², a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ²(''n''₀, ''s''₀²) can be a convenient form to represent a more informative prior for σ², 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 \; \exp \left -\frac \right/math>
Such a prior would lead to the posterior distribution
:p(\sigma^2, D, I^\prime, \mu) \propto \frac \; \exp \left -\frac \right/math>
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
p(\mu, \sigma^2 \mid D, I) & \propto \frac \exp \left -\frac \right\\
& = \frac \exp \left -\frac \right\exp \left -\frac \right\end

The marginal posterior distribution for σ² is obtained from the joint posterior distribution by integrating out over μ,
:\begin
p(\sigma^2, D, I) \; \propto \; & \frac \; \exp \left -\frac \right\; \int_^ \exp \left -\frac \rightd\mu\\
= \; & \frac \; \exp \left -\frac \right\; \sqrt \\
\propto \; & (\sigma^2)^ \; \exp \left -\frac \right\end
This is again a scaled inverse chi-squared distribution, with parameters $\scriptstyle\;$ and $\scriptstyle$.

Related distributions

* If $X \sim \mbox\chi^2(\nu, \tau^2)$ then $k X \sim \mbox\chi^2(\nu, k \tau^2)\,$
* If $X \sim \mbox\chi^2(\nu) \,$ (Inverse-chi-squared distribution) then $X \sim \mbox\chi^2(\nu, 1/\nu) \,$
* If $X \sim \mbox\chi^2(\nu, \tau^2)$ then $\frac \sim \mbox\chi^2(\nu) \,$ (Inverse-chi-squared distribution)
* If $X \sim \mbox\chi^2(\nu, \tau^2)$ then $X \sim \textrm\left(\frac, \frac\right)$ (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

{{DEFAULTSORT:Scaled-Inverse-Chi-Squared Distribution
Continuous distributions
Exponential family distributions