Inverse-gamma distribution

Characterization

Probability density function

The inverse gamma distribution's probability density function is defined over the support x 0

: f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} (1/x)^{\alpha + 1}\exp\left(-\beta/x\right)

with shape parameter \alpha and scale parameter \beta. Here \Gamma(\cdot) denotes the gamma function.

Unlike the gamma distribution, which contains a somewhat similar exponential term, \beta is a scale parameter as the density function satisfies: : f(x; \alpha, \beta) = \frac{f(x / \beta; \alpha, 1)}{\beta}

Cumulative distribution function

The cumulative distribution function is the regularized gamma function

:F(x; \alpha, \beta) = \frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(\alpha)} = Q\left(\alpha, \frac{\beta}{x}\right)!

where the numerator is the upper incomplete gamma function and the denominator is the gamma function. Many math packages allow direct computation of Q, the regularized gamma function.

Moments

Provided that \alpha n, the n-th moment of the inverse gamma distribution is given by

:\mathrm{E}[X^n] = \beta^n \frac{\Gamma(\alpha - n)}{\Gamma(\alpha)} = \frac{\beta^n}{(\alpha - 1) \cdots (\alpha - n)}.

Characteristic function

The inverse gamma distribution has characteristic function \frac{2\left(-i\beta t\right)^{!!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right) where K_\alpha is the modified Bessel function of the 2nd kind.

Properties

For \alpha0 and \beta0, : \mathbb{E}[\ln(X)] = \ln(\beta) - \psi(\alpha), and : \mathbb{E}[X^{-1}] = \frac{\alpha}{\beta},,

The information entropy is

: \begin{align} \operatorname{H}(X) & = \operatorname{E}[-\ln(p(X))] \ & = \operatorname{E}\left[-\alpha \ln(\beta) + \ln(\Gamma(\alpha)) + (\alpha+1)\ln(X) + \frac{\beta}{X}\right] \ & = -\alpha \ln(\beta) + \ln(\Gamma(\alpha)) + (\alpha+1)\ln(\beta) - (\alpha+1)\psi(\alpha) + \alpha\ & = \alpha + \ln(\beta\Gamma(\alpha)) - (\alpha+1)\psi(\alpha). \end{align}

where \psi(\alpha) is the digamma function.

The Kullback-Leibler divergence of Inverse-Gamma(αp, βp) from Inverse-Gamma(αq, βq) is the same as the KL-divergence of Gamma(αp, βp) from Gamma(αq, βq):

D_{\mathrm{KL}}(\alpha_p,\beta_p; \alpha_q, \beta_q) = \mathbb{E}\left[ \log \frac{\rho(X)}{\pi(X)}\right] = \mathbb{E}\left[ \log \frac{\rho(1/Y)}{\pi(1/Y)}\right] = \mathbb{E}\left[ \log \frac{\rho_G(Y)}{\pi_G(Y)}\right],

where \rho, \pi are the pdfs of the Inverse-Gamma distributions and \rho_G, \pi_G are the pdfs of the Gamma distributions, Y is Gamma(αp, βp) distributed.

: \begin{align} D_{\mathrm{KL}}(\alpha_p,\beta_p; \alpha_q, \beta_q) = {} & (\alpha_p-\alpha_q) \psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) + \alpha_q(\log \beta_p - \log \beta_q) + \alpha_p\frac{\beta_q-\beta_p}{\beta_p}. \end{align}

Related distributions

• If X \sim \mbox{Inv-Gamma}(\alpha, \beta) then k X \sim \mbox{Inv-Gamma}(\alpha, k \beta) ,, for k 0
• If X \sim \mbox{Inv-Gamma}(\alpha, \tfrac{1}{2}) then X \sim \mbox{Inv-}\chi^2(2 \alpha), (inverse-chi-squared distribution)
• If X \sim \mbox{Inv-Gamma}(\tfrac{\alpha}{2}, \tfrac{1}{2}) then X \sim \mbox{Scaled Inv-}\chi^2(\alpha,\tfrac{1}{\alpha}), (scaled-inverse-chi-squared distribution)
• If X \sim \textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2}) then X \sim \textrm{Levy}(0,c), (Lévy distribution)
• If X \sim \textrm{Inv-Gamma}(1,c) then \tfrac{1}{X} \sim \textrm{Exp}(c), (Exponential distribution)
• If X \sim \mbox{Gamma}(\alpha, \beta), (Gamma distribution with rate parameter \beta) then \tfrac{1}{X} \sim \mbox{Inv-Gamma}(\alpha, \beta), (see derivation in the next paragraph for details)
• Note that If X \sim \mbox{Gamma}(k, \theta) (Gamma distribution with scale parameter \theta ) then 1/X \sim \mbox{Inv-Gamma}(k, 1/\theta)
• Inverse gamma distribution is a special case of type 5 Pearson distribution
• A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.
• For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)

Derivation from Gamma distribution

Let X \sim \mbox{Gamma}(\alpha, \beta), and recall that the pdf of the gamma distribution is

: f_{X}(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, x 0.

Note that \beta is the rate parameter from the perspective of the gamma distribution.

Define the transformation Y = g(X) = \tfrac{1}{X}. Then, the pdf of Y is

:\begin{align} f_Y(y) &= f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right| \[6pt] &= \frac{\beta^\alpha}{\Gamma(\alpha)} \left( \frac{1}{y} \right)^{\alpha-1} \exp \left( \frac{-\beta}{y} \right) \frac{1}{y^2} \[6pt] &= \frac{\beta^\alpha}{\Gamma(\alpha)} \left( \frac{1}{y} \right)^{\alpha+1} \exp \left( \frac{-\beta}{y} \right) \[6pt] &= \frac{\beta^\alpha}{\Gamma(\alpha)} y^{-\alpha-1} \exp \left( \frac{-\beta}{y} \right) \[6pt] \end{align}

Note that {\beta} is the scale parameter from the perspective of the inverse gamma distribution. This can be straightforwardly demonstrated by seeing that {\beta} satisfies the conditions for being a scale parameter. :\begin{align} \frac{f(y / \beta; \alpha, 1)}{\beta} &= \frac{1}{\beta} \frac{1}{\Gamma(\alpha)} \left( \frac{y}{\beta} \right)^{-\alpha-1} \exp \left(-\frac{1}{y / \beta}\right) \[6pt] &= \frac{\beta^\alpha}{\Gamma(\alpha)} y^{-\alpha-1} \exp \left(-\frac{\beta}{y}\right) \[6pt] &= f(y; \alpha, \beta) \end{align}

Occurrence

• Hitting time distribution of a Wiener process follows a Lévy distribution, which is a special case of the inverse-gamma distribution with \alpha=0.5.

References

References

1. Hoff, P.. (2009). "A First Course in Bayesian Statistical Methods". Springer.
2. "InverseGammaDistribution—Wolfram Language Documentation".
3. John D. Cook. (Oct 3, 2008). "InverseGammaDistribution".
4. Ludkovski, Mike. (2007). "Math 526: Brownian Motion Notes". UC Santa Barbara.