- Story. If \(x\) is Gamma distributed, then \(1/x\) is Inverse Gamma distributed.
- Parameters. The number of arrivals, \(α\), and the rate of arrivals, \(β\).
- Support. The Inverse Gamma distribution is supported on the set of positive real numbers.
- Probability density function.
\(\begin{align}
f(y;\alpha, \beta) = \frac{1}{\Gamma(\alpha)}\,\frac{\beta^\alpha}{y^{(\alpha+1)}}\,\mathrm{e}^{-\beta/ y}
\end{align}\)
- Usage
| Package |
Syntax |
| NumPy |
1 / np.random.gamma(alpha, 1/beta) |
| SciPy |
`scipy.stats.invgamma(alpha, loc=0, scale=beta) |
| Stan |
inv_gamma(alpha, beta) |
- Notes.
- The Inverse Gamma distribution is useful as a prior for positive parmeters. It imparts a quite heavy tail and keeps probability further from zero than the Gamma distribution.
- The
numpy.random 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 as shown in the NumPy usage above.
params = [dict(name='α', start=0.01, end=2, value=0.5, step=0.01),
dict(name='β', start=0.1, end=2, value=1, step=0.01)]
app = distribution_plot_app(x_min=0,
x_max=20,
scipy_dist=st.invgamma,
params=params,
transform=lambda alpha, beta: (alpha, 0, beta),
x_axis_label='y',
title='Inverse Gamma')
bokeh.io.show(app, notebook_url=notebook_url)