Markov Chain Monte Carlo sampling for conditional tests: A link between permutation tests and algebraic statistics

We consider conditional tests for non-negative discrete exponential families. We develop two Markov Chain Monte Carlo (MCMC) algorithms which allow us to sample from the conditional space and to perform approximated tests. The first algorithm is based on the MCMC sampling described by Sturmfels. The second MCMC sampling consists in a more efficient algorithm which exploits the optimal partition of the conditional space into orbits of permutations. We thus establish a link between standard permutation and algebraic-statistics-based sampling. Through a simulation study we compare the exact cumulative distribution function (cdf) with the approximated cdfs which are obtained with the two MCMC samplings and the standard permutation sampling. We conclude that the MCMC sampling which exploits the partition of the conditional space into orbits of permutations gives an estimated cdf, under $H_0$, which is more reliable and converges to the exact cdf with the least steps. This sampling technique can also be used to build an approximation of the exact cdf when its exact computation is computationally infeasible.