Inverse logistic function

Definition

The inverse logistic function or log-odds function is a function from the open interval $(0,1)$ to all of $\mathbb{R}$ defined as follows:

$x\mapsto \ln (\frac{x}{1-x})$

The function may be extended to a function $[0,1]\to [-\mathrm{\infty },\mathrm{\infty }]$ with the value at 0 defined as $-\mathrm{\infty }$ and the value at 1 defined as $\mathrm{\infty }$.

Probabilistic interpretation

Given a probability $p$ (strictly between 0 and 1) the inverse logistic function computes the logarithm of the corresponding odds. Explicitly, the odds corresponding to probability $p$ are:

$\frac{p}{1-p}$

The logarithm of the odds is therefore:

$\ln (\frac{p}{1-p})$

Key data

Item	Value
default domain	open interval $(0,1)$
range	all of $\mathbb{R}$
inverse function	logistic function $x\mapsto \frac{1}{1+e^{-x}}$