Scaled inverse chi-squared distribution

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Scaled inverse chi-squared
Probability density function
Scaled inverse chi squared.svg
Cumulative distribution function
Scaled inverse chi squared cdf.svg
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[edit]

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[edit]

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[edit]

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 L2|D) = p(D2) 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 (xi-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[edit]

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[edit]

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[edit]

References[edit]

  • 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