Probability-based classification

A probabilistic classifier outputs $\hat{p}=P(y=1\mid \mathbf{x})$, and you turn it into a decision with a threshold $\tau$: predict positive if $\hat{p}\ge \tau$. 0.5 is the default, but not always the best.

Change the threshold to reflect costs. For cancer screening, a missed case is far worse than a false alarm, so lower $\tau$ to catch more positives (higher recall, lower precision). For spam, flagging a real email is costly, so raise $\tau$.

Calibration means the probabilities are honest: of all emails given 0.8, about 80% really are spam. Logistic regression is usually well calibrated; trees, SVMs and deep networks often are not and can be fixed with Platt scaling or isotonic regression.

The loss used for probabilities is cross-entropy: $L=-[y\log \hat{p}+(1-y)\log (1-\hat{p})]$. It rewards confidence when right and punishes confident mistakes severely.