The scaled inverse chisquared 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/σ^{2} = τ^{2}. The distribution is therefore parametrised by the two quantities ν and τ^{2}, referred to as the number of chisquared degrees of freedom and the scaling parameter, respectively.
This family of scaled inverse chisquared distributions is closely related to two other distribution families, those of the inversechisquared distribution and the inversegamma distribution. Compared to the inversechisquared distribution, the scaled distribution has an extra parameter τ^{2}, which scales the distribution horizontally and vertically, representing the inversevariance of the original underlying process. Also, the scaled inverse chisquared 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
 then
Compared to the inverse gamma distribution, the scaled inverse chisquared distribution describes the same data distribution, but using a different parametrization, which may be more convenient in some circumstances. Specifically, if
 then
Either form may be used to represent the maximum entropy distribution for a fixed first inverse moment and first logarithmic moment .
The scaled inverse chisquared 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 chisquared 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 σ_{0}^{2} rather than by τ^{2}, and has a different interpretation. The application has been more usually presented using the inversegamma distribution formulation instead; however, some authors, following in particular Gelman et al. (1995/2004) argue that the inverse chisquared parametrisation is more intuitive.
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Transcription
Contents
Characterization
The probability density function of the scaled inverse chisquared distribution extends over the domain and is
where is the degrees of freedom parameter and is the scale parameter. The cumulative distribution function is
where is the incomplete gamma function, is the gamma function and is a regularized gamma function. The characteristic function is
where is the modified Bessel function of the second kind.
Parameter estimation
The maximum likelihood estimate of is
The maximum likelihood estimate of can be found using Newton's method on:
where is the digamma function. An initial estimate can be found by taking the formula for mean and solving it for Let be the sample mean. Then an initial estimate for is given by:
Bayesian estimation of the variance of a Normal distribution
The scaled inverse chisquared 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:
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(Dσ^{2}) has the familiar form
Combining this with the rescalinginvariant 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
This form can be recognised as that of a scaled inverse chisquared distribution, with parameters ν = n and τ^{2} = s^{2} = (1/n) Σ (x_{i}μ)^{2}
Gelman et al remark that the reappearance 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 rescalinginvariant 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}:
In the samplingtheory case, conditioned on σ^{2}, the probability distribution for (1/s^{2}) is a scaled inverse chisquared distribution; and so the probability distribution for σ^{2} conditioned on s^{2}, given a scaleagnostic prior, is also a scaled inverse chisquared distribution.
Use as an informative prior
If more is known about the possible values of σ^{2}, a distribution from the scaled inverse chisquared family, such as Scaleinvχ^{2}(n_{0}, s_{0}^{2}) can be a convenient form to represent a less uninformative prior for σ^{2}, as if from the result of n_{0} previous observations (though n_{0} need not necessarily be a whole number):
Such a prior would lead to the posterior distribution
which is itself a scaled inverse chisquared distribution. The scaled inverse chisquared 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 translationinvariant prior p(μI) ∝ const., which gives the following joint posterior distribution for μ and σ^{2},
The marginal posterior distribution for σ^{2} is obtained from the joint posterior distribution by integrating out over μ,
This is again a scaled inverse chisquared distribution, with parameters and .
Related distributions
 If then
 If (Inversechisquared distribution) then
 If then (Inversechisquared distribution)
 If then (Inversegamma 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
 ^ Gelman et al (1995), Bayesian Data Analysis (1st ed), p.68