 Solution

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

μ

p (D) = \frac{p ( D ∣ μ ) p ( μ )}{p ( μ ∣ D )} .

(2 π)^{- d /2} ∣Σ ∣^{- 1/2}

at its mean. Therefore

p (D) ≃ p (D ∣ μ) p (μ) (2 π)^{d /2} ∣Σ ∣^{1/2} .

Equivalently, the approximate log marginal likelihood is

lo g p (D) ≃ lo g p (D ∣ μ) + lo g p (μ) + \frac{d}{2} lo g (2 π) + \frac{1}{2} lo g ∣Σ∣ .

The determinant factor accounts for the volume of the locally plausible parameter region, which the peak likelihood alone does not measure.

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