Sampling from a multivariate Gaussian distribution truncated on a simplex: A review

In many Bayesian models, the posterior distribution of interest is a multivariate Gaussian distribution restricted to a specific domain. In particular, when the unknown parameters to be estimated can be considered as proportions or probabilities, they must satisfy positivity and sum-to-one constraints. This paper reviews recent Monte Carlo methods for sampling from multivariate Gaussian distributions restricted to the standard simplex. First, a classical Gibbs sampler is presented. Then, two Hamiltonian Monte Carlo methods are described and analyzed. In a similar fashion to the Gibbs sampler, the first method has a acceptance rate equal to one whereas the second requires an accept/reject procedure. The performance of the three methods are compared through the use of a few examples.