Logistic function

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

More generally, there exists a family of logistic functions that can be written as $f\left(x\right)=\frac{M}{1+{\alpha }^{c(x_0-x)}}$, 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.