Contents

Inverse Gamma¶

It is the inverse of a Gamma distribution:

\[\begin{split} Y \sim IG(a,b) \\ Y = \frac{1}{X} \\ X \sim Ga(a,b \end{split}\]

PDF¶

\[ IG(x| a, b) \triangleq \frac{b^a}{\Gamma(a)} x^{- (a+ 1)}e ^{- b/x} \]

Moments¶

Mean¶

\(E[X] = \frac{b}{a -1}\)

Exists only if \(a > 1\)

Variance¶

\(var[X] = \frac{b^2}{(a - 1) ^2 (a - 2)}\)

Exists only if \(a > 2\)

Mode¶

\(mode(X) = \frac{b}{a +1}\)

Numerically stable ridge regression computation Conjugate gradient