Optimal Scaling for Random walk Metropolis on spherically constrained target densities

We consider the problem of optimal scaling of the proposal variance for multidimensional Random walk Metropolis (RWM) algorithms. It is well known, for a wide range of continuous target densities, that the optimal scaling of the proposal variance leads to an average acceptance rate of 0.234. Therefore a natural question is, do similar results for target densities which have discontinuities? In the current work, we answer in the affirmative for a class of spherically constrained target densities. Even though the acceptance probability is more complicated than for continuous target densities, the optimal scaling of the proposal variance again leads to an average acceptance rate of 0.234