How to sample from a truncated distribution if you must

Sampling from a truncated distribution is difficult. There are currently two major methods proposed for solving this task. The first proposed solution is a random-walk MCMC algorithm. Although it eventually gives the correct distribution, it can be very slow in multi-modal distributions. The second approach called the ellipsoid method is practically more efficient for problems in which users have good prior information, but a correctness is not guaranteed. In this paper, we present a framework which can unify these two approaches. The key idea is to merge both methods into a single Markov chain using a trick called Metropolis-coupled MCMC. Once merged, they can validly exchange information to each other. Although the chain constructed from the ellipsoid approach cannot be proven to be correct, it usually rapidly converges to a useful stationary distribution, and its information can help the other chain constructed by the random-walk approach to converge faster to the correct distribution.