2005 International Conference on Analysis of Algorithms
Conrado Martínez (ed.)
DMTCS Conference Volume AD (2005), pp. 371382
author:  Shuji Kijima and Tomomi Matsui 

title:  Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex 
keywords:  Markov chain, Mixing time, Path coupling, Coupling from the past, Logconcave function. 
abstract: 
In this paper, we are concerned with random sampling of an
n
dimensional integral point on an
(n1)
dimensional simplex according to a multivariate
discrete distribution. We employ sampling via Markov chain
and propose two ``hitandrun'' 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
, namely
2
ln(Kɛ
1
))
(1/2)n(n1) ln(K ɛ
, where
1
)
K
is the side length of the simplex and
ɛ
(
0<ɛ<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 is bounded by
O(n
.
3
ln(Kn))

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reference:  Shuji Kijima and Tomomi Matsui (2005), Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex , in 2005 International Conference on Analysis of Algorithms, Conrado Martínez (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AD, pp. 371382 
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