principles.fyi · the brain · concept

sigmoid

The S-curve that translates a score on the log-odds ruler back into a probability.

sigmoid(z) = 1 / (1 + e^-z)

The logistic sigmoid 1/(1+e^-z) is the inverse of the log-odds map: if a score z is the log-odds of an event, then sigmoid(z) is the event's probability. It is not a curve anyone chose for its looks — it is the only function consistent with 'this score is a log-odds', and its S-shape is exactly the squash of the probability ruler: huge score changes far from zero barely move p, because out there the ruler is crushed. It converts summed evidence into probability in logistic regression, score gaps into preference probabilities in Bradley-Terry reward models, and is softmax's two-option special case.