Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 4 / b / ii

Table of contents
- Solution ii

Solution

For

q_{ϕ} = N (μ_{Q}, σ_{Q}^{2})

,

ELBO (μ_{Q}, σ_{Q}^{2}) = - \frac{1}{2} lo g (2 π σ^{2}) - \frac{( y - μ _{Q} ) ^{2} + σ _{Q}^{2}}{2 σ ^{2}} - \frac{1}{2} [lo g \frac{τ ^{2}}{σ _{Q}^{2}} + \frac{σ _{Q}^{2} + μ _{Q}^{2}}{τ ^{2}} - 1] .

Differentiation gives

μ_{Q}^{*} = \frac{τ ^{2}}{σ ^{2} + τ ^{2}} y = θ, (σ_{Q}^{2})^{*} = \frac{σ ^{2} τ ^{2}}{σ ^{2} + τ ^{2}} = σ_{θ}^{2} .

The Gaussian variational family contains the exact posterior, so its best member is the posterior itself and the maximized ELBO equals

lo g Z

.

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