Probability Distributions and their Stories

• Story. The Half-Normal distribution is a Gaussian distribution with zero mean truncated to only have nonzero probability for positive real numbers.
• Parameters. Strictly speaking, the Half-Normal is parametrized by a single positive parameter $σ$. We could, however, translate it so that the truncated Gaussian has a maximum at $μ$ and only values greater than or equal to $μ$ have nonzero probability.
• Probability density function.
  $\begin{align}f(y;\mu, \sigma) = \left\{\begin{array}{ccc}\frac{2}{\sqrt{2\pi \sigma^2}}\,\mathrm{e}^{-(y-\mu)^2/2\sigma^2} & & y \ge \mu\\0 && \text{otherwise}\end{array}\right.\end{align}$
  
  
• Support. The Half-Normal distribution is supported on the set $[μ,∞)$. Usually, we have $μ=0$, in which case the Half-Normal distribution is supported on the set of nonnegative real numbers.
• Usage
  
  

  Package	Syntax
  NumPy	np.random.normal(mu, sigma)
  SciPy	scipy.stats.halfnorm(mu, sigma)
  Stan	real<lower=mu> y; y ~ normal(mu, sigma)

  
  
• Notes.
  • In Stan, a Half-Normal is defined by putting bounds on the variable and then using a Normal distribution.
  • The Half-Normal distibution is a useful prior for nonnegative parameters that should not be too large and may be very close to zero.

params = [dict(name='µ', start=-0.5, end=0.5, value=0, step=0.01),
          dict(name='σ', start=0.1, end=1.0, value=0.2, step=0.01)]
app = distribution_plot_app(x_min=-0.5,
                            x_max=2,
                            scipy_dist=st.halfnorm,
                            params=params, 
                            x_axis_label='y',
                            title='Half-Normal')
bokeh.io.show(app, notebook_url=notebook_url)