Inverse logistic function

Definition

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

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

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

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:

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

Key data

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