Inverse Gamma distribution¶
Story¶
If \(Y\) is Gamma distributed, then \(1/Y\) is Inverse Gamma distributed.
Parameters¶
The number of arrivals, \(\alpha\), and the rate of arrivals, \(\beta\).
Support¶
The Inverse Gamma distribution is supported on the set of positive real numbers.
Probability density function¶
Moments¶
Mean: \(\displaystyle{\frac{\beta}{\alpha  1}}\) for \(\alpha > 1\); for \(\alpha \le 1\), the mean is undefined.
Variance: \(\displaystyle{\frac{\beta^2}{(\alpha1)^2(\alpha2)}}\) for \(\alpha > 2\); for \(\alpha \le 2\), the variance is undefined.
Usage¶
Package 
Syntax 

NumPy 

SciPy 

Stan 

Notes¶
The Inverse Gamma distribution is useful as a prior for positive parameters. It imparts a quite heavy tail and keeps probability further from zero than the Gamma distribution.
NumPy module does not have a function to sample directly from the Inverse Gamma distribution, but it can be achieved by sampling out of a Gamma distribution and then taking the inverser, as shown in the NumPy usage above.