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

Table of contents
- Solution e

Solution

For normalized target density

p (m ∣ d)

and proposal

q

, the unbiased importance estimator

K^{- 1} \sum_{i} m_{i} p (m_{i} ∣ d) / q (m_{i})

\frac{1}{K} [\int \frac{m ^{2} p ( m ∣ d ) ^{2}}{q ( m )} d m - E [m ∣ d]^{2}] .

By Cauchy-Schwarz inequality,

\int \frac{m ^{2} p ( m ∣ d ) ^{2}}{q ( m )} d m \geq (\int ∣ m ∣ p (m ∣ d) d m)^{2},

with equality exactly when

q^{*} (m) = \frac{∣ m ∣ p ( m ∣ d )}{\int ∣ m ∣ p ( m ∣ d ) d m} .

This is rarely useful because constructing and sampling from it already requires detailed knowledge of the posterior and its absolute first moment, the objects importance sampling was meant to avoid computing.

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