 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/15/b/solution

Include an intercept

w_{0}

and let

z

be the chosen numerical feature vector. Logistic regression uses

h (z; w) = σ (w_{0} + w^{T} z), σ (u) = \frac{1}{1 + e ^{- u}} .

The logistic function maps every real score into

(0, 1)

, allowing a Bernoulli distribution model for the label. Equivalently, the log odds are linear in the features:

lo g \frac{h}{1 - h} = w_{0} + w^{T} z .

This gives a smooth differentiable probability model with a linear decision boundary at any fixed probability threshold. The sigmoid output is a fitted conditional probability, not a categorical prediction until a threshold is chosen.

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