Inverse logistic function

From Calculus
Jump to: navigation, search

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}}