InverseGamma

Probability density on $x > 0$:

$$p(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\!\left(-\frac{\beta}{x}\right)$$

Moments (when defined):

$$\mathbb{E}[X] = \frac{\beta}{\alpha - 1}\;\;(\alpha > 1), \qquad \mathrm{Var}[X] = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)} \;\;(\alpha > 2)$$

Mode: $\beta / (\alpha + 1)$.

Conjugacy: with a $\mathrm{Normal}(\mu, \sigma^2)$ likelihood and known mean $\mu$, the posterior over $\sigma^2$ is again Inverse-Gamma with updated parameters $\alpha' = \alpha + n/2$, $\beta' = \beta + \tfrac{1}{2}\sum_i (x_i - \mu)^2$.

Because Lucid's lucid.distributions.Gamma sampler is rejection-based, has_rsample is False here and sample returns detached samples obtained by reciprocating a Gamma draw.