  We present the simplicial sampler, a class of parallel MCMC methods that
generate and choose from multiple proposals at each iteration. The algorithm's
multiproposal randomly rotates a simplex connected to the current Markov chain
state in a way that inherently preserves symmetry between proposals. As a
result, the simplicial sampler leads to a simplified acceptance step: it simply
chooses from among the simplex nodes with probability proportional to their
target density values. We also investigate a multivariate Gaussian-based
symmetric multiproposal mechanism and prove that it also enjoys the same
simplified acceptance step. This insight leads to significant theoretical and
practical speedups. While both algorithms enjoy natural parallelizability, we
show that conventional implementations are sufficient to confer efficiency
gains across an array of dimensions and a number of target distributions.
