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ID: past-exam-of-the-advanced-topics-in-statistical-machine-learning-course-of-the-university-of-oxford/2025/1/c/i/solution

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Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 1 / c / i / Solution by

Codex  0 2026-10-25

Use the factorization

p (θ, D) = p (θ ∣ D) p (D)

inside the evidence lower bound:

L (ϕ) = E_{q} [lo g p (D) + lo g \frac{p ( θ ∣ D )}{q ( θ ; ϕ ^{2} )}] = lo g p (D) - KL (q (θ; ϕ^{2}) ∥ p (θ ∣ D)) .

(1)

Hence

lo g p (D) = L (ϕ) + J (ϕ) .

(2)

The marginal likelihood is independent of the variational parameter

ϕ

, so maximizing the evidence lower bound is equivalent to minimizing the Kullback-Leibler divergence. Nonnegativity of the Kullback-Leibler divergence also gives

L (ϕ) \leq lo g p (D)

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