Decision Boundary / Thresholding

The problem, from first principles

Sigmoid produces a continuous confidence, but an application often needs a discrete class. Thresholding turns every probability into the label convention used by the rest of the classifier while allowing a caller to choose a stricter or looser cutoff.

From theory to code

Implement predict_labels(p, threshold=0.5) with one vectorized comparison. Theory states which side of the boundary includes an exact tie.

Constraints

• p may be any array shape; return labels of exactly that shape.
• Return integer 0 and 1 values, not a Boolean array.
• Use threshold=0.5 by default and honor a supplied threshold.
• A probability exactly equal to the threshold is positive.
• Do not use loops or modify p.

Hints

Open one at a time. Each gives away a little more than the last.