Logistic function

[summary: The logistic function is a [-sigmoid] function that maps the real numbers to the unit interval $(0, 1)$ using the formula $\displaystyle f(x) = \frac{1}{1 + e^{-x}}$ or variants of this formula.]

The logistic function is a [-sigmoid] function that maps the real numbers to the unit interval $(0, 1)$ using the formula $\displaystyle f(x) = \frac{1}{1 + e^{-x}}$.

More generally, there exists a [family_of_functions family] of logistic functions that can be written as $\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}$, where:

• $M$ is the upper bound of the function (in which case the function maps to the interval $(0, M)$ instead). When $M = 1$, the logistic function is usually being used to calculate some Probability or Proportion of a total.

• $x_0$ is the inflection point of the curve, or the value of $x$ where the function's growth stops speeding up and starts slowing down.

• $\alpha$ is a variable controlling the steepness of the curve.

• $c$ is a scaling factor for the distance.

Applications

• The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.

• The inverse logistic function (with $\alpha = 2$) is used to convert a probability to log-odds form for use in Bayes' rule.

• The logistic function (with $\alpha = 10$ and $c = 1/400$) is used to calculate the expected probability of a player winning given a specific difference in rating in the Elo rating system.