Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 40 / 5 / c / Solution

Use the random-walk proposal specified by its centre and covariance: for the current state $z=(x,y)$ draw $Z_1,Z_2$ independently standard normal and setIts properly normalized density of the proposal distribution isIt is symmetric: $q(z^{\prime }\mid z)=q(z\mid z^{\prime })$. This is a centred-at-the-current-state proposal, rather than an unshifted draw from the density notation in the source. The random-walk Metropolis algorithm consequently needs only the target ratio. Put $H(x,y)=x^2+y^2-2\rho xy$. ThenAn independent uniform decides the move; on rejection both coordinates remain at their previous values. A proposal with lower $H$ is always accepted. This construction uses the joint density directly, without needing either conditional distribution. Its continuous positive proposal density supplies communication over the target support, and the holding probability avoids a deterministic periodic transition pattern.