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((short description|Two-parameter family of continuous probability distributions))
((refimprove|date=October 2014))
((Probability distribution|
  name       =Inverse-gamma|
  type       =density|
  pdf_image  =[[Image:Inv gamma pdf.svg|325px]]|
  cdf_image  =[[Image:Inv gamma cdf.svg|325px]]|
  parameters =<math>\alpha>0</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\beta>0</math> [[scale parameter|scale]] ([[real number|real]])|
  support    =<math>x\in(0,\infty)\!</math>|
  pdf        =<math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(-\frac{\beta}{x}\right)</math>|
  cdf        =<math>\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!</math>|
  mean       =<math>\frac{\beta }{\alpha-1}\!</math> for <math>\alpha > 1</math>|
  median     =|
  mode       =<math>\frac{\beta}{\alpha+1}\!</math>|
  variance   =<math>\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!</math> for <math>\alpha > 2</math>|
  skewness   =<math>\frac{4\sqrt{\alpha-2)){\alpha-3}\!</math> for <math>\alpha > 3</math>|
  kurtosis   =<math>\frac{6(5\,\alpha-11)}{(\alpha-3)(\alpha-4)}\!</math> for <math>\alpha > 4</math>|
  entropy    =<math>\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\psi(\alpha)</math>
<br />(see [[digamma function]])|
  mgf        =Does not exist.|
  char       =<math>\frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2))}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)</math>|
))

In [[probability theory]] and [[statistics]], the '''inverse gamma distribution''' is a two-parameter family of continuous [[probability distribution]]s on the positive [[real line]], which is the distribution of the [[multiplicative inverse|reciprocal]] of a variable distributed according to the [[gamma distribution]].  

Perhaps the chief use of the inverse gamma distribution is in [[Bayesian statistics]], where the distribution arises as the marginal posterior distribution for the unknown [[variance]] of a [[normal distribution]], if an [[uninformative prior]] is used, and as an analytically tractable [[conjugate prior]], if an informative prior is required. It is common among some Bayesians to consider an alternative [[Statistical parameter|parametrization]] of the [[normal distribution]] in terms of the [[precision (statistics)|precision]], defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior.  Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a [[scaled inverse chi-squared distribution]].

==Characterization==

===Probability density function===
The inverse gamma distribution's [[probability density function]] is defined over the [[support (mathematics)|support]] <math>x > 0</math>

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

with [[shape parameter]] <math>\alpha</math> and <!-- see the definition of scale for distributions!!! --> [[scale parameter]] <math>\beta</math>.<ref>((cite web|url=http://reference.wolfram.com/language/ref/InverseGammaDistribution.html|title=InverseGammaDistribution—Wolfram Language Documentation|website=reference.wolfram.com|access-date=9 April 2018))</ref>  Here <math>\Gamma(\cdot)</math> denotes the [[gamma function]].

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

===Cumulative distribution function===
The [[cumulative distribution function]] is the [[Incomplete gamma function#Regularized Gamma functions and Poisson random variables|regularized gamma function]]

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

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

===Moments===
Provided that <math>\alpha > n</math>, the <math>n</math>-th moment of the inverse gamma distribution is given by<ref>((cite web|url=https://www.johndcook.com/inverse_gamma.pdf|title=InverseGammaDistribution|author=John D. Cook|date= Oct 3, 2008|access-date=3 Dec 2018))</ref> 

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

===Characteristic function===
The inverse gamma distribution has [[characteristic function (probability theory)|characteristic function]] <math display=block>\frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2))}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)</math> where <math>K_\alpha</math> is the [[modified Bessel function]] of the 2nd kind.

==Properties==
For <math>\alpha>0 </math> and <math>\beta>0</math>,
: <math>\mathbb{E}[\ln(X)] = \ln(\beta) - \psi(\alpha)\, </math>
and
: <math>\mathbb{E}[X^{-1}] = \frac{\alpha}{\beta},\, </math>

The [[information entropy]] is

: <math>
\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}
</math>

where  <math>\psi(\alpha) </math> is the [[digamma function]].

The [[Kullback–Leibler divergence|Kullback-Leibler divergence]] of Inverse-Gamma(''α<sub>p</sub>'', ''β<sub>p</sub>'') from Inverse-Gamma(''α<sub>q</sub>'', β<sub>''q''</sub>) is the same as the KL-divergence of Gamma(''α<sub>p</sub>'', ''β<sub>p</sub>'') from Gamma(''α<sub>q</sub>'', ''β<sub>q</sub>''):

<math>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], </math>

where  <math>\rho, \pi </math> are the pdfs of the Inverse-Gamma distributions and  <math>\rho_G, \pi_G </math> are the pdfs of the Gamma distributions, <math>Y </math> is Gamma(''α<sub>p</sub>'', ''β<sub>p</sub>'') distributed.

: <math>
\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}
</math>

==Related distributions==
* If <math>X \sim \mbox{Inv-Gamma}(\alpha, \beta)</math> then <math> k X \sim \mbox{Inv-Gamma}(\alpha, k \beta) \,</math>, for <math> k > 0 </math>
* If <math>X \sim \mbox{Inv-Gamma}(\alpha, \tfrac{1}{2})</math> then <math>X \sim \mbox{Inv-}\chi^2(2 \alpha)\,</math> ([[inverse-chi-squared distribution]])
* If <math>X \sim \mbox{Inv-Gamma}(\tfrac{\alpha}{2}, \tfrac{1}{2})</math> then <math>X \sim \mbox{Scaled Inv-}\chi^2(\alpha,\tfrac{1}{\alpha})\,</math> ([[scaled-inverse-chi-squared distribution]])
* If <math>X \sim \textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})</math> then <math>X \sim \textrm{Levy}(0,c)\,</math> ([[Lévy distribution]])
* If <math>X \sim \textrm{Inv-Gamma}(1,c)</math> then <math>\tfrac{1}{X} \sim \textrm{Exp}(c)\,</math> ([[Exponential distribution]])
* If <math>X \sim \mbox{Gamma}(\alpha, \beta)\,</math> ([[Gamma distribution]] with ''rate'' parameter <math>\beta</math>) then <math>\tfrac{1}{X} \sim \mbox{Inv-Gamma}(\alpha, \beta)\,</math> (see derivation in the next paragraph for details) 
*Note that If <math>X \sim \mbox{Gamma}(k, \theta)</math> (Gamma distribution with scale parameter <math>\theta</math> ) then <math>1/X \sim \mbox{Inv-Gamma}(k, 1/\theta)</math> 
* Inverse gamma distribution is a special case of type 5 [[Pearson distribution]]
* A [[multivariate random variable|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 <math>X \sim \mbox{Gamma}(\alpha, \beta)</math>, and recall that the pdf of the [[gamma distribution]] is

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

Note that <math> \beta </math> is the rate parameter from the perspective of the gamma distribution.

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

:<math>\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)} \left( y \right)^{-\alpha-1} \exp \left(  \frac{-\beta}{y}  \right) \\[6pt]
\end{align}</math>

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


==Occurrence==

* [[Hitting time]] distribution of a [[Wiener process]] follows a [[Lévy distribution]], which is a special case of the inverse-gamma distribution with <math>\alpha=0.5</math>.<ref>((Cite web|last=Ludkovski|first=Mike|date=2007|title=Math 526: Brownian Motion Notes|url=http://ludkovski.faculty.pstat.ucsb.edu/bmNotes.pdf|publisher=UC Santa Barbara|pages=5–6))</ref>

==See also==
*[[Gamma distribution]]
*[[Inverse-chi-squared distribution]]
*[[Normal distribution]]
*  [[Pearson distribution]]

==References==
((Reflist))
*Hoff, P. (2009). "A first course in bayesian statistical methods". Springer.
*((cite journal |first=V. |last=Witkovsky |year=2001 |title=Computing the Distribution of a Linear Combination of Inverted Gamma Variables |journal=Kybernetika |volume=37 |issue=1 |pages=79–90 |zbl=1263.62022 |mr=1825758 ))

((ProbDistributions|continuous-semi-infinite))

((DEFAULTSORT:Inverse-Gamma Distribution))
[[Category:Continuous distributions]]
[[Category:Conjugate prior distributions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Exponential family distributions]]
[[Category:Gamma and related functions]]

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