Scores and Boundaries

The Sigmoid Link

The sigmoid link introduces the named-boundary register. Only the midpoint is exact; the other sigmoid entries stay symbolic.

highlighted = computed this step

The sigmoid link

The sigmoid maps a score into the open unit interval. Its formula contains an exponential, so this is the first named boundary.

σ(z)=11+e−z\sigma(z)=\frac{1}{1+e^{-z}}
Named sigmoid columnThe probability column names sigmoid except at the exact midpoint.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero

The one exact midpoint

When z=0, sigmoid equals 1/2 exactly. The other sigmoid cells are named symbols, not decimal values.

σ(0)=1/2\sigma(0)=1/2
Named sigmoid columnThe probability column names sigmoid except at the exact midpoint.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero

Summary

Sigmoid and logarithm are named boundary operations here; the rendered structure is exact, but no decimal value is pinned for them. The threshold decision from the previous lesson did not need sigmoid.

named boundary: σ\text{named boundary: }\sigma
Named sigmoid columnThe probability column names sigmoid except at the exact midpoint.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero