OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 5K / Solution

Logistic regression assumes independent

Y_{i} ∣ X_{i} \sim Bernoulli (π_{i})

with

lo g \frac{π _{i}}{1 - π _{i}} = β_{0} + X_{i}^{T} β .

For the binomial exponential family,

ℓ = y θ - m lo g (1 + e^{θ}) + constant

, so the mean is

m e^{θ} / (1 + e^{θ})

and the canonical parameter is its logit. It may be fitted by iteratively reweighted least squares.

Under LDA, Bayes' rule gives

lo g \frac{P ( Y = 1 ∣ x )}{P ( Y = 0 ∣ x )} = l o g \frac{π _{1}}{π _{0}} + (μ_{1} - μ_{0})^{T} Σ^{- 1} x - f r a c 12 (μ_{1}^{T} Σ^{- 1} μ_{1} - μ_{0}^{T} Σ^{- 1} μ_{0}),

which is a logistic model with an intercept.

Solved by gpt-5.6-sol high.

 Ancestors (10)