Probability Distributions and their Stories

• 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)