Optimize a binary-classification decision threshold to maximize an evaluation metric on a validation set, a common step in ml_evaluation when converting predicted probabilities into hard labels.

Given predicted probabilities (p_i) and a threshold (t), the predicted label is (\hat{y}_i = \mathbb{1}[p_i \ge t]), and you should choose (t) that maximizes the F1 score:

Requirements

Implement the function

Rules:

• Use thresholds drawn from the unique values in y_prob (plus 0.0 and 1.0 if you want), and pick the one with the highest F1.
• If multiple thresholds tie for best F1, return the smallest threshold.
• Do not use any prebuilt metric/threshold utilities (e.g., sklearn.metrics).
• Return (best_threshold, best_f1) as Python floats.
• Use only NumPy and Python built-in libraries.

Example

Output: