The sigmoid shape
The sigmoid maps any real number to the range zero to one with an S shape. Large positive inputs approach one, large negative inputs approach zero, and an input of zero maps exactly to one half.
Where the boundary lives
A classifier with a default threshold of one half predicts class one when the probability exceeds one half. That happens precisely when the linear score is positive. So the decision boundary is the surface where the score equals zero.
It is linear
- For logistic regression the boundary is a straight line, plane, or hyperplane.
- The model can only separate classes a linear surface can split.
- Curved boundaries require feature transforms or nonlinear models.
Moving the boundary
- Lowering the threshold below one half enlarges the predicted positive region.
- Raising it shrinks that region, trading recall for precision.
- The boundary location is a deployment choice, not fixed by training.
Key idea
The sigmoid turns scores into probabilities, and the default decision boundary sits where the score is zero. For logistic regression that boundary is a flat hyperplane.