Probability Distributions and their Stories

• Story. Say you wait for two multistep processes to happen. The individual steps of each process happen at the same rate, but the first multistep process requires $α$ steps and the second requires $β$ steps. The fraction of the total waiting time take by the first process is Beta distributed.
• Example.
• Parameters. There are two parameters, both strictly positive: $α$ and $β$, defined in the above story.
• Support. The Beta distribution has support on the interval [0, 1].
• Probability density function.
  
  $\begin{align}f(\theta;\alpha, \beta) = \frac{\theta^{\alpha-1}(1-\theta)^{\beta-1}}{B(\alpha,\beta)}\end{align}\$
  
  where
  
  $\begin{align}B(\alpha,\beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha + \beta)}\end{align}$
  
  is the Beta function.
• Usage
  
  

  Package	Syntax
  NumPy	np.random.beta(alpha, beta)
  SciPy	scipy.stats.beta(alpha, beta)
  Stan	weibull(alpha, sigma)

  
  
  
• Related distributions.
  • The Uniform distribution on the interval [0, 1] is a special case of the Beta distribution with $α=β=1$.
• Notes.
  • The story of the Beta distribution is difficult to parse. Most importantly for our purposes, the Beta distribution allows us to put probabilities on unknown probabilities. It is only defined on $0≤x≤1$, and $x$ here can be interpreted as a probability, say of a Bernoulli trial.
  • The case where $a=b=0$ is not technically a probability distribution because the PDF cannot be normalized. Nonetheless, it can be used as an improper prior, and this prior is known a Haldane prior, names after biologist J. B. S. Haldane. The case where $a=b=1/2$ is sometimes called a Jeffreys prior.
    

params = [dict(name='α', start=0.01, end=10, value=1, step=0.01),
          dict(name='β', start=0.01, end=10, value=1, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=1,
                            scipy_dist=st.beta,
                            params=params,
                            x_axis_label='θ',
                            title='Beta')
bokeh.io.show(app, notebook_url=notebook_url)