Scores and Boundaries

The Decision Threshold

The threshold uses the sign of the exact score. The boundary is where the score is zero, and the diagram shows that point directly.

highlighted = computed this step

The threshold rule

Class 1 is chosen exactly when z is at least 0; otherwise class 0 is chosen.

z≥0⇒1z\ge{}0\Rightarrow1
Decision thresholdThe exact z sign and the z equals zero boundary determine the class decision.Classifier score tablexzdecision0-201-10201311Decision boundaryThe exact point where the score is zero.ABCDz=0

Why the boundary is exact

The boundary is the point where z=0. Which side a point is on is determined by the sign of z.

z=0 is the boundaryz=0\text{ is the boundary}
Decision thresholdThe exact z sign and the z equals zero boundary determine the class decision.Classifier score tablexzdecision0-201-10201311Decision boundaryThe exact point where the score is zero.ABCDz=0

The decision on a number line

The exact scores -2, -1, 0, and 1 sit on the score axis. The cut at z=0 sends class 0 left and class 1 right, giving the same decisions 0, 0, 1, 1 the table already shows.

z≥0⇒class 1z\ge{}0\Rightarrow\text{class }1
The decision on a number lineEach exact score is placed on the score axis; the cut at z=0 splits the classes.threshold t=0 on the exact score axisclass 0 (z below 0)class 1 (z ≥ 0)-3-2-1012threshold z=0z=-2z=-1z=0z=1the score axis and the cut at z=0 are exact (σ(0)=1/2 exactly, the only exactpoint of the named sigmoid curve); the threshold is a STATED decision policyplaced by the author, NOT a learned boundary: this does not claim the cut isfitted, NOT a probability, NOT a generalization claim

Summary

The decisions 0, 0, 1, and 1 match the shown labels. No sigmoid value was needed for this exact decision.

sign(z) decides the class\text{sign}(z)\text{ decides the class}
Decision thresholdThe exact z sign and the z equals zero boundary determine the class decision.Classifier score tablexzdecision0-201-10201311Decision boundaryThe exact point where the score is zero.ABCDz=0