Logistic Normal Distribution

probability-theory

Logistic-Normal Distribution

A logistic-normal distribution is a continuous distribution on the open interval $x\in (0,1)$ obtained by applying the logistic (sigmoid) map to a normal variable:

$$\begin{array}{rl}z& \sim \mathcal{N}(\mu ,{\omega }^2)\\ s& =\mathrm{sigmoid}(z)=\frac{1}{1+e^{-}z}\in (0,1)\end{array}$$

Probability Density Function

Let $\mathrm{logit}(x)=\ln \left(\frac{x}{1-x}\right)$. The probability density function is defined as:

$$P(x)=f_X(x;\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }\cdot x(1-x)}\exp \left(-\frac{(logit(x)-\mu {)}^2}{2{\sigma }^2}\right),0<x<1$$

Distribution Function

Let $\Phi$ be the standard normal distribution function. The distribution function is given by:

$$D(x)=F_x(x;\mu ,\sigma )=\Phi \left(\frac{\mathrm{logit}(x)-\mu }{\sigma }\right),0<x<1$$