Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 (Astrostatistics)

A spherical shell contributes a factor $4\pi r^2$ to the number of stars. Since ${\int }_0^{\infty }{r}^2{e}^{-r/r_0}dr=2r_0^3$, the normalized distance density isThus $C=(2r_0^3{)}^{-1}$, $\alpha =2$, and $\beta =1$. This is a gamma distribution with shape three and scale $r_0$.

Ignoring terms independent of $r_0$, the log-likelihood isIts score vanishes atThe expected Fisher information isBecause a shape-three gamma variable has mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound ${\mathcal{I}}_N(r_0{)}^{-1}$.

The inverse-square law gives $f_s=L_0/(4\pi r_s^2)$, so a star is observed exactly whenWriting $a=R/r_0$, integration of the shape-three gamma density givesTherefore the fully normalized truncated distribution is

At distance $r_s$, detection requires $L_s\ge 4\pi r_s^2{f}_{\min }$. HenceThe probability is $1/2$ when its normal quantile is zero, namely at

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.

Stack $y=(y_1^{\top },y_2^{\top }{)}^{\top }$. After integrating out the Ornstein-Uhlenbeck process,Writing $k(a,b)=A^2{e}^{-|a-b|/\tau }$, the covariance blocks areand $({\Sigma }_{12}{)}_{jk}=k(t_j,t_k-\Delta t)$, with ${\Sigma }_{21}={\Sigma }_{12}^{\top }$. Therefore

With $D=(y_1,y_2,t)$ and $\nu =(\Delta m,c,A,\tau )$,A Random-walk Metropolis algorithm proposes ${\theta }^{\prime }=\theta +Z$ from a symmetric multivariate normal increment, conveniently using logarithmic coordinates for $A,\tau >0$, and accepts with probabilityAfter burn-in, retain a suitably long chain and assess convergence and effective sample size of a Markov chain.

Let $\pi$ denote the posterior and let the proposal density satisfy $q({\theta }^{\prime }\mid \theta )=q(\theta \mid {\theta }^{\prime })$. For distinct states, the Metropolis transition density isConsequentlyThe rejection mass on the diagonal also satisfies detailed balance, so the posterior is invariant.

Discard burn-in from the MCMC output and retain the sampled $\Delta t$ coordinate. A normalized histogram or kernel density estimation of these draws approximates $p(\Delta t\mid D)$. Autocorrelation changes the Monte Carlo uncertainty, so uncertainty bands should use the chain's effective sample size rather than its raw length.

The light-curve posterior used the analysis prior ${\pi }_0(\Delta t)=N(0,{\sigma }_{\mathrm{prior}}^2)$, so its marginal likelihood as a function of delay is proportional to $p(\Delta t\mid D)/{\pi }_0(\Delta t)$. ThereforeThe integral can be evaluated numerically using the approximation from part (d).

With equal model prior probabilities, Bayesian model averaging gives the unnormalized densityNormalizing this expression over the allowed interval automatically incorporates each lens model's evidence and hence its posterior model probability.

Normal-normal conjugacy givesThe three quantities in the proposed identity areand, by the Kullback-Leibler divergence between normal distributions,Substitution and simplification show that the latter two expressions differ by exactly $\log Z$, so the equality holds.

Bayes theorem givesTaking its posterior expectation after subtracting $\log \pi (\theta )$ yieldsThus the equality holds for every proper prior and valid likelihood for which the displayed expectations are well-defined.

For any approximating density $q_{\varphi }$,The evidence lower bound is thereforeSince $\log Z$ does not depend on $\varphi$, maximizing the ELBO is equivalent to minimizing the divergence from $q_{\varphi }$ to the posterior.

For $q_{\varphi }=N({\mu }_Q,{\sigma }_Q^2)$,Differentiation givesThe Gaussian variational family contains the exact posterior, so its best member is the posterior itself and the maximized ELBO equals $\log Z$.