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DISTRIBUTIONS / CONTINUOUS / INVERSE GAMMA

Inverse Gamma distribution

What it represents

The Inverse Gamma distribution models a positive variable whose reciprocal is Gamma. Its right tail is heavy; alpha determines which moments exist and beta sets inverse scale.

Historical clue

It is the reciprocal transform of Gamma and became established in Bayesian inference as a conjugate distribution for a normal variance under scale or precision parameterizations.

Relationships that clarify its use

If X is Gamma, 1/X is Inverse Gamma. It is related to Beta prime and F through Gamma and chi-square ratios; the meaning of beta depends on whether the model describes scale or precision.

Data examples

  • priors for unknown variances and scales
  • lifetimes and positive phenomena with very large possible values

Modelling warning

Check that the moment of interest exists before interpreting a posterior mean or using a quadratic loss.

A Gamma law on the other side of a reciprocal

If a precision or rate is Gamma, its reciprocal is Inverse Gamma. This construction explains the distribution’s historical role as a prior for positive variances and scales. Probability may concentrate away from zero while a power-law tail allows very large values.

That tail imposes clear conditions. The mean exists only when alpha exceeds one, and variance requires a value above two. Fitting the density without checking these limits can produce a numerical summary that the model does not define. In Bayesian work, conjugate convenience does not guarantee a sensible prior either: a seemingly diffuse choice may place unexpected mass near zero or on enormous scales.

Decision guide

A good candidate when: a positive quantity is the reciprocal of a Gamma variable or represents a heavy-tailed scale or variance.

Compare it with: Lognormal and Pareto. Check which moments exist for the estimated shape; mean or variance may not be valid summaries.

References

Inverse Gamma Distribution: equations and calculator

Distribution defintion

X∼InverseGamma(α,β)X\sim\mathrm{InverseGamma}\left(\alpha,\beta\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

α∈R+,β∈R+\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−γ(α,β/x)Γ(α)=1−P(α,βx)F_{X}\left(x\right)=1-\frac{\gamma(\alpha,\beta/x)}{\Gamma(\alpha)}=1-\text{P}\left(\alpha,\frac{\beta}{x}\right)

Probability density function

fX(x)=βαΓ(α)x−α−1exp⁡(−βx)f_{X}\left(x\right)=\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha-1} \exp\left(-\frac{\beta}{x}\right)

Percent point function/Sample

FX−1(u)=βP−1(α,1−u)F^{-1}_{X}\left(u\right)=\frac{\beta}{\text{P}^{-1}\left(\alpha,1-u\right)}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=Γ(α−k)Γ(α)=1(α−1)⋯(α−k)if α>k\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{\Gamma(\alpha-k)}{\Gamma(\alpha)}=\frac{1}{(\alpha-1) \cdots (\alpha-k)}\quad \text{if } \alpha>k

Parametric mean

Mean(X)=βμ~1′\mathrm{Mean}(X)=\beta\tilde{\mu}'_{1}

Parametric variance

Variance(X)=β2(μ~2′−μ~1′2)\mathrm{Variance}(X)=\beta^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}

Parametric median

Median(X)=βP−1(α,12)\mathrm{Median}(X)=\frac{\beta}{\text{P}^{-1}\left(\alpha,\frac{1}{2}\right)}

Parametric mode

Mode(X)=βα+1\mathrm{Mode}(X)=\frac{\beta}{\alpha+1}

Additional information and definitions

X~∼InverseGamma(α,1)\tilde{X}\sim\mathrm{InverseGamma}\left(\alpha,1\right)
β:Scale parameter\beta:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}