Logistic probability and loss

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Problem

A binary logistic model has logit $z=w_1x_1+w_2x_2+b$ . State its feature-space decision boundary at threshold $0.5$ . Then, for an example with $z=\ln 4$ and $y=0$ , compute $P(y=1\mid x)$ , the predicted class, and binary cross-entropy.

Reveal answer or reference solution

Since $\sigma(z)=0.5$ exactly when $z=0$ , the decision boundary is

w_1x_1+w_2x_2+b=0.

Also, $\sigma(\ln 4)=\frac{4}{5}$ . Since $\frac{4}{5}\ge 0.5$ , the predicted class is $1$ . For $y=0$ , the loss is

-\ln\left(1-\frac{4}{5}\right)=\ln 5\approx 1.6094379124.

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Logistic probability and loss

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