Probability Distributions and their Stories

• Story. If \(ln y\) is Gaussian distributed, \(y\) is Log-Normally distributed.
• Example. A measure of fold change in gene expression can be Log-Normally distributed.
• Parameters. As for a Gaussian, there are two parameters, the mean, \(μ\), and the variance \(σ^2\).
• Support. The Log-Normal distribution is supported on the set of positive real numbers.
• Probability density function.
  
  
  \(\begin{align}
  f(y;\mu, \sigma) = \frac{1}{y\sqrt{2\pi \sigma^2}}\,\mathrm{e}^{-(\ln y - \mu)^2/2\sigma^2}
  \end{align}\)
  
  
  
• Usage
  
  

Package	Syntax
NumPy	np.random.lognormal(mu, sigma)
SciPy	scipy.stats.lognorm(x, sigma, loc=0, scale=np.exp(mu))
Stan	lognormal(mu, sigma)

• Notes.
• Be careful not to get confused. The Log-Normal distribution describes the distribution of \(y\) given that \(ln y\) is Gaussian distributed. It does not describe the distribution of \(ln y\).
  • The way location, scale, and shape parameters work in SciPy for the Log-Normal distribution is confusing. If you want to specify a Log-Normal distribution as we have defined it using scipy.stats, use a shape parameter equal to \(σ\), a location parameter of zero, and a scale parameter given by \(e^μ\). For example, to compute the PDF, you would use scipy.stats.lognorm(x, sigma, loc=0, scale=np.exp(mu)).
  • The definition of the Log_Normal in the numpy.random module matches what we have defined above and what is defined in Stan.

params = [dict(name='µ', start=0.01, end=0.5, value=0.1, 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,
                            x_max=4,
                            scipy_dist=st.lognorm,
                            params=params, 
                            transform=lambda mu, sigma: (sigma, 0, np.exp(mu)),
                            x_axis_label='y',
                            title='Log-Normal')

bokeh.io.show(app, notebook_url=notebook_url)