principles.fyi · the brain · concept

log-odds

The straightened belief ruler: log of the odds, symmetric around 0 and unbounded both ways.

log-odds = log( p / (1 - p) )

Log-odds (the 'logit') is log(p/(1-p)) — probability with the squash removed in two moves: odds remove the ceiling, the log removes the lopsidedness. Perfectly unsure (50%) sits at exactly 0, belief and disbelief mirror each other as +x and -x, and equal steps mean equal multiplications of the odds everywhere on the ruler. That makes it the scale where evidence adds, where logistic regression sums its feature scores, and where the Bradley-Terry model places its score difference; the sigmoid is simply the trip back from this ruler to probability.