Inverse-gamma distribution

Inverse-gamma
	Probability density function
	Cumulative distribution function
Parameters	shape (real); scale (real)
Support
PDF
CDF
Mean	for
Mode
Variance	for
Skewness	for
Ex. kurtosis	for
Entropy	; (see digamma function)
MGF	Does not exist.
CF

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the 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.

However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the 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 $x>0$

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 $\beta$ .^[1] 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 distribution 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.

Characteristic function

$K_{\alpha }(\cdot )$ in the expression of the characteristic function is the modified Bessel function of the 2nd kind.

Properties

For $\alpha >0$ and $\beta >0$ ,

\mathbb {E} [\ln(X)]=\ln(\beta )-\psi (\alpha )\,

and

\mathbb {E} [X^{-1}]={\frac {\alpha }{\beta }},\,

The information entropy is

{\begin{aligned}\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{aligned}}

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

The Kullback-Leibler divergence of Inverse-Gamma(α_p, β_p) from Inverse-Gamma(α_q, β_q) is the same as the KL-divergence of Gamma(α_p, β_p) from Gamma(α_q, β_q):

$D_{\mathrm {KL} }(\alpha _{p},\beta _{p};\alpha _{q},\beta _{q})=\mathbb {E} \left[\log {\frac {\rho (X)}{\pi (X)}}\right]=\mathbb {E} \left[\log {\frac {\rho (1/Y)}{\pi (1/Y)}}\right]=\mathbb {E} \left[\log {\frac {\rho _{G}(Y)}{\pi _{G}(Y)}}\right],$

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(α_p, β_p) distributed.

Related distributions

If $X\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )$ then $kX\sim {\mbox{Inv-Gamma}}(\alpha ,k\beta )\,$
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 $X \sim \mbox{Gamma}(\alpha, \beta)\,$ (Gamma distribution) then $\tfrac{1}{X} \sim \mbox{Inv-Gamma}(\alpha, \beta)\,$ (see derivation in the next paragraph for details)
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

The pdf of the gamma distribution is

f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}

and define the transformation $Y=g(X)={\tfrac {1}{X}}$ then the resulting transformation is

{\begin{aligned}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{aligned}}

References

↑ "InverseGammaDistribution—Wolfram Language Documentation". reference.wolfram.com. Retrieved 9 April 2018.

V. Witkovsky (2001) Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37(1), 79-90

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "InverseGammaDistribution—Wolfram Language Documentation". reference.wolfram.com. Retrieved 9 April 2018.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped