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:

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

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 $\mathbb {R}$
inverse function	logistic function $x\mapsto {\frac {1}{1+e^{-x}}}$