Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 3

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

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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).

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.