BCE — clip the probabilities so log stays finite

Background

Binary cross-entropy is

$$L_{\text{BCE}} = -\frac{1}{n}\sum_i \bigl[y_i\log\hat{y}_i + (1-y_i)\log(1-\hat{y}_i)\bigr].$$

The catch: $\log(0) = -\infty$. If a model ever predicts exactly $0$ or $1$, the naive formula returns nan/inf. Real implementations clip $\hat{y}$ into $[\varepsilon, 1-\varepsilon]$ first so the log stays finite.

Problem statement

Implement clipped_bce(y, y_hat, eps=1e-7) that clips the predictions to $[\varepsilon, 1-\varepsilon]$ and returns the mean BCE.

Input

• y — array-like of labels (0/1).
• y_hat — array-like of predicted probabilities in $[0, 1]$ (may include exactly 0 or 1).
• eps — float, the clip margin (default 1e-7).

Output

Returns a Python float: a finite mean BCE.

Examples

Example 1 — ordinary case

Input:  y = [1, 0], y_hat = [0.9, 0.1]
Output: 0.1053605...

Explanation: $-\tfrac12(\log 0.9 + \log 0.9)$.

Example 2 — exact 0/1 no longer explodes

Input:  y = [1, 0], y_hat = [1.0, 0.0]
Output: ~1.2e-7  (finite, tiny)

Explanation: clipping to $1-\varepsilon$ / $\varepsilon$ keeps the log finite instead of nan.

Constraints

• Clip y_hat to [eps, 1 - eps] before taking logs.
• Average over the $n$ examples.
• Return a finite float.

Notes

• A confident wrong prediction (e.g. $\hat{y}=\varepsilon$ when $y=1$) still produces a large loss $\approx -\log\varepsilon$ — the clip bounds it instead of letting it become infinite.