Logistic probability and cross-entropy

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Problem

A binary logistic model produces logit $z=\ln(3)$ for a positive example $y=1$ . Compute $p(y=1\mid\mathbf{x})$ , the predicted class at threshold $0.5$ , and binary cross-entropy loss.

Reveal answer or reference solution

$\sigma(\ln 3)=\frac{3}{4},$

so the predicted class is $1$ , and the loss is

$-\ln\left(\frac{3}{4}\right)=\ln\left(\frac{4}{3}\right)\approx 0.28768.$

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