Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques.

Abstract

We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication in the Iwahori-Hecke algebra.persi diaconis and arun ram is the normalizing constant. Thus π(w) is smallest when w = id and, as θ → 1, π tends to the uniform distribution. These non-uniform distributions arise in statistical work as Mallows models. Background and references are in Section 2e.A standard Monte Carlo Markov chain algorithm for sampling from π is the Metropolis algorithm with a systematic scan. This algorithm cycles through the generators in order. If multiplying by the current generator increases length this multiplication is made. If the length decreases, then the multiplication is made with probability θ and omitted with probability 1 − θ. One scan uses s , in order. Define K(w, w ′ ) = the chance that a systematic scan started at w ends in w ′ .(1.2)Repeated scans of the algorithm are defined byIn Section 2c and 4a we show that this Markov chain has π as unique stationary distribution.The main results of this paper derive sharp results on rates of convergence for these walks. As an example of what our methods give, we show that order n scans are necessary and suffice to reach stationary on the symmetric group starting from the identity. More precisely, we prove

Mixing times of the biased card shuffling and the asymmetric exclusion process

“…This verifies a conjecture of Diaconis and Ram [7]. A lower bound for the mixing time of the form N 2 is easy and well known.…”

“…However, the probability space contains elements of very small probability, so the term log(1/ min x π(x)) is of order N 2 (see [7], where the stationary distribution for the Metropolis chain is given). Thus (5) yields a bound of order N 2 .…”

“…[10] or [7]). In [7] Diaconis and Ram studied a different version of biased card shuffling. In their model the selection of the pair of adjacent cards was not random, but done in a prescribed deterministic manner ("systematic scan").…”

Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

“…The literature on Mallows and Generalized Mallows models ranges from theoretical discussion [15], [17], [19], [24], [42], [43] to more practical applications to multilabel classification [9], label ranking [8] or estimation of distribution algorithms [7].…”

Some Hecke algebra products and corresponding random walks

“…Note. A probabilistic interpretation of certain Hecke algebra products different from ours appears in a paper by Diaconis and Ram [4].…”