Binary output head — probability and decision

Background

For binary classification the network sets o = 1, so the single sigmoid output $a_2 \in (0,1)$ reads directly as $\mathbb{P}(\text{class }1)$. From that one number you get everything the head needs:

$$\mathbb{P}(1) = a_2, \qquad \mathbb{P}(0) = 1 - a_2, \qquad \hat{y} = \begin{cases}1 & a_2 > 0.5\\ 0 & a_2 \le 0.5\end{cases}$$

Problem statement

Implement binary_head(a2) returning the tuple (p0, p1, label) for a single scalar output activation.

Input

• a2 — float in $(0, 1)$, the sigmoid output (probability of class 1).

Output

Returns (p0, p1, label): p0 = 1 - a2 (float), p1 = a2 (float), label = 1 if a2 > 0.5 else 0 (int).

Examples

Example 1

Input:  a2 = 0.7
Output: (0.3, 0.7, 1)

Example 2 — exactly 0.5 decides class 0

Input:  a2 = 0.5
Output: (0.5, 0.5, 0)

Explanation: the rule is $a_2 > 0.5$ (strict), so a tie goes to class 0.

Constraints

• p0 = 1 - a2, p1 = a2, both floats.
• label = int(a2 > 0.5).
• Return them in the order (p0, p1, label).

Notes

• The 0.5 threshold is the neutral default; shifting it trades precision against recall without retraining the network.