Logistic function

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 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.