Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 2 / c

Table of contents
- Solution c

Solution

If equally informative independent draws have variance

s_{m}^{2}

, an ordinary mean of

K_{eff}

draws has variance

s_{m}^{2} / K_{eff}

. The weighted mean has the corresponding variance

s_{m}^{2} \sum_{i} w_{i}^{2}

. Equating them gives the effective sample size of importance sampling

K_{eff} = \frac{1}{\sum _{i = 1}^{K} w _{i}^{2}} = \frac{( \sum _{i} p ( d ∣ x _{i} ) ) ^{2}}{\sum _{i} p ( d ∣ x _{i} ) ^{2}} \leq K,

where the inequality follows from Cauchy-Schwarz inequality.

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