Logit-Normal Distribution#

The logit-normal distribution is a two-parameter continuous probability distribution. A logit-normal random variable is a variable whose logit is a normally distributed random variable.

Logit and logistic function

The logit function is the inverse cumulative distribution function of the standard logistic distribution. It is defined as

\[ \mathrm{logit}(x) = \ln \frac{x}{1 - x}, \; x \in (0, 1). \]

The inverse of the logit function is called the logistic function (it’s, the cumulative distribution function of the standard logistic distribution):

\[ \mathrm{logistic}(x) = \frac{1}{1 + e^{-x}}, \; x \in \mathbb{R} \]

The range of the logistic function is \((0, 1)\).

The table below summarizes some important aspects of the distribution.

Notation

\(X \sim \mathcal{N}_{\mathrm{logit}}(\mu, \sigma)\)

Parameters

\(\mu \in \mathbb{R}\)

\(\sigma > 0\)

Support

\(\mathcal{D}_X = (0, 1)\)

PDF

\(f_X (x; \mu, \sigma) = \begin{cases} 0.0 & x \leq 0.0 \\ \frac{1}{\sigma \sqrt{2 \pi}} \exp{\left[ - \frac{1}{2} \left(\frac{\mathrm{logit}(x) - \mu}{\sigma} \right)^2 \right]} & x \in (0, 1) \\ 0.0 & x \geq 1.0 \end{cases}\)

CDF

\(F_X (x; \mu, \sigma) = \begin{cases} 0.0 & x \leq 0.0 \\ \frac{1}{2} \left[ 1 + \mathrm{erf}\left( \frac{\mathrm{logit}(x) - \mu}{\sigma \sqrt{2}}\right) \right] & x \in (0, 1) \\ 1.0 & x \geq 1.0\\ \end{cases}\)

ICDF

\(F^{-1}_X (x; \mu, \sigma) = \mathrm{logistic} \left(\mu + \sqrt{2} \, \sigma \, \mathrm{erf}^{-1}(2 x - 1) \right)\)

Note

The parameters \(\mu\) and \(\sigma\) of a logit-normal distribution correspond to the mean and standard deviation of the underlying normal distribution, respectively.

The plots of probability density functions (PDFs), sample histogram (of \(5'000\) points), cumulative distribution functions (CDFs), and inverse cumulative distribution functions (ICDFs) for different parameter values are shown below.

../../_images/logitnormal_2_0.png