 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-219/4/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

p (θ ∣ y) = N (θ, σ_{θ}^{2}), σ_{θ}^{2} = \frac{σ ^{2} τ ^{2}}{σ ^{2} + τ ^{2}}, θ = \frac{τ ^{2}}{σ ^{2} + τ ^{2}} y .

The three quantities in the proposed identity are

lo g Z = - \frac{1}{2} [lo g (2 π (σ^{2} + τ^{2})) + \frac{y ^{2}}{σ ^{2} + τ ^{2}}],

E_{θ ∣ y} lo g L (θ) = - \frac{1}{2} lo g (2 π σ^{2}) - \frac{( y - θ ) ^{2} + σ _{θ}^{2}}{2 σ ^{2}},

D_{KL} (p (θ ∣ y) ∥ π) = \frac{1}{2} [lo g \frac{τ ^{2}}{σ _{θ}^{2}} + \frac{σ _{θ}^{2} + θ ^{2}}{τ ^{2}} - 1] .

Substitution and simplification show that the latter two expressions differ by exactly

lo g Z

, so the equality holds.

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