In this paper, we propose a perfect (exact) sampling algorithm according to a discretized Dirichlet distribution. Our algorithm is a monotone coupling from the past algorithm, which is a Las Vegas type randomized algorithm. We propose a new Markov chain whose limit distribution is a discretized Dirichlet distribution. Our algorithm simulates transitions of the chain O(n ln #) times where n is the dimension (the number of parameters) and 1/# is the grid size for discretization. Thus the obtained bound does not depend on the magnitudes of parameters. In each transition, we need to sample a random variable according to a discretized beta distribution (2-dimensional Dirichlet distribution). To show the polynomiality, we employ the path coupling method carefully and show that our chain is rapidly mixing.

Abstract not found

In this paper, we are concerned with random sampling of an n dimensional integral point on an (n − 1) dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two “hit-and-run ” chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in O(n 2 ln(Kε −1)), namely (1/2)n(n − 1)ln(Kε −1), where K is the side length of the simplex and ε (0 < ε < 1) is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler