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

The likelihood isUsing the conditional independence $d\perp m\mid x$ and the joint prior $p(x,m)$,and

The simulation catalog is a sample from the joint prior. Self-normalized importance sampling therefore estimates the posterior mean byThe likelihood weights update each simulated system according to how well its satellite resembles the measured one.

If equally informative independent draws have variance $s_m^2$, an ordinary mean of $K_{\mathrm{eff}}$ draws has variance $s_m^2/K_{\mathrm{eff}}$. The weighted mean has the corresponding variance $s_m^2{\sum }_i{w}_i^2$. Equating them gives the effective sample size of importance samplingwhere the inequality follows from Cauchy-Schwarz inequality.

The stated conditional independences giveThusThe dependence among satellite properties and host mass remains encoded in each jointly simulated row.

For normalized target density $p(m\mid d)$ and proposal $q$, the unbiased importance estimator $K^{-1}{\sum }_i{m}_{i}p(m_i\mid d)/q(m_i)$ has varianceBy Cauchy-Schwarz inequality,with equality exactly whenThis 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.