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# InverseGamma

The **Inverse Gamma** distribution on $(0, \infty)$ with shape $\alpha > 0$ and
rate $\beta > 0$:

$$
f(x \mid \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}\,
x^{-\alpha-1} e^{-\beta/x}, \qquad x > 0.
$$

If $X \sim \mathrm{Gamma}(\alpha, \beta)$ then $1/X \sim
\mathrm{InvGamma}(\alpha, \beta)$. It is the $a \to 0$ limit of the
{doc}`GIG <gig>` and the subordinator behind the {doc}`NormalInverseGamma
<normal_inverse_gamma>` mixture. The right tail is polynomial, so the mean
exists only for $\alpha > 1$ and the variance only for $\alpha > 2$.

## Density gallery

Smaller $\alpha$ makes the right tail heavier; the mode sits at
$\beta/(\alpha+1)$, well below the mean.

```{code-cell} python
:tags: [hide-input]
import jax
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt

from normix import InverseGamma
from normix.utils.plotting import set_theme

set_theme()
np.set_printoptions(precision=5, suppress=True)

x = jnp.linspace(1e-3, 5.0, 400)
fig, ax = plt.subplots()
for alpha in [1.5, 2.0, 3.0, 5.0]:
    g = InverseGamma(alpha=jnp.array(alpha), beta=jnp.array(1.0))
    ax.plot(np.asarray(x), np.asarray(jax.vmap(g.pdf)(x)), label=rf"$\alpha$ = {alpha:g}")
ax.set_xlabel("x"); ax.set_ylabel("density")
ax.set_title(r"InverseGamma densities ($\beta$ = 1)")
ax.legend()
plt.show()
```

## Parametrizations

Stored attributes: `alpha`, `beta`. Exponential family with sufficient
statistic $t(x) = [-1/x,\; \log x]$:

| Parametrization | Value |
|---|---|
| Classical | shape $\alpha > 0$, rate $\beta > 0$ |
| Natural $\theta$ | $[\,\beta,\; -(\alpha + 1)\,]$ |
| Expectation $\eta = \nabla\psi$ | $[\,-\alpha/\beta,\; \log\beta - \psi(\alpha)\,] = (\mathbb{E}[-1/X],\; \mathbb{E}[\log X])$ |

where $\psi$ is the digamma function.

## Quick usage

```{code-cell} python
ig = InverseGamma(alpha=jnp.array(3.0), beta=jnp.array(2.0))

print("mean/var/std:", float(ig.mean()), float(ig.var()), float(ig.std()))
print("cdf(1.0):    ", float(ig.cdf(jnp.array(1.0))))
print("95% quantile:", float(ig.ppf(jnp.array(0.95))))

samples = ig.rvs(10_000, seed=0)
fitted = InverseGamma.fit_mle(samples)     # moment-matching MLE
print("fit (a,b):   ", float(fitted.alpha), float(fitted.beta))
```

## See also

- API: {py:class}`~normix.distributions.inverse_gamma.InverseGamma`
- Theory: {doc}`../theory/gig` — InverseGamma as a limit of the GIG family
- Tutorial: {doc}`../tutorials/distributions/01_univariate_positive`