Inverse-gamma distribution

The inverse-gamma distribution with shape $a>0$ and scale $b>0$ has probability density functionEquivalently, $Y=1/X$ has a gamma distribution with shape $a$ and rate $b$. The inverse change of variables has absolute Jacobian determinant $x^{-2}$, which turns the gamma density into the displayed expression. Thus the inverse-gamma scale parameter is the reciprocal variable's gamma rate, not its gamma scale.

Its moments satisfy $\mathbb{E}{X}^r=b^r\Gamma (a-r)/\Gamma (a)$ when $r<a$. For positive $r\ge a$ the defining integral diverges at infinity. In particular,For $a=3,b=2$, the expected value and variance are both one. The distribution is useful for positive scale uncertainty, but its heavy tail means that a valid normalized probability distribution can still lack a finite expected value.

Let $X_1,\dots ,X_n$ be conditionally independent inverse-gamma distribution observations with known shape $k>0$, unknown scale $\Theta >0$, and observed values $x_i>0$. Their likelihood function isA gamma distribution prior with shape $\alpha >0$ and rate $\lambda >0$ is a conjugate prior, givingThis follows by multiplying the two powers of $\Theta$ and the two exponential factors. It uses reciprocal observations, rather than their sample mean.

If $k>1$, the conditional claim expected value is $m(\Theta )=\Theta /(k-1)$. Its Bayes estimator under squared error loss is the posterior expected valueHere the target is the conditional expected value, whose posterior second moment is finite because the posterior for $\Theta$ is gamma. When $0<k\le 1$, the conditional claim expected value is infinite and this finite-estimate formula does not apply. For example, $k=2$, $\alpha =1$, $\lambda =1$ and observations $1,2$ give posterior shape $5$, rate $5/2$, and estimated conditional claim expected value $2$.