1,064 research outputs found
Speeding up the FMMR perfect sampling algorithm: A case study revisited
In a previous paper by the second author,two Markov chain Monte Carlo perfect
sampling algorithms -- one called coupling from the past (CFTP) and the other
(FMMR) based on rejection sampling -- are compared using as a case study the
move-to-front (MTF) self-organizing list chain. Here we revisit that case study
and, in particular, exploit the dependence of FMMR on the user-chosen initial
state. We give a stochastic monotonicity result for the running time of FMMR
applied to MTF and thus identify the initial state that gives the
stochastically smallest running time; by contrast, the initial state used in
the previous study gives the stochastically largest running time. By changing
from worst choice to best choice of initial state we achieve remarkable speedup
of FMMR for MTF; for example, we reduce the running time (as measured in Markov
chain steps) from exponential in the length n of the list nearly down to n when
the items in the list are requested according to a geometric distribution. For
this same example, the running time for CFTP grows exponentially in n.Comment: 19 pages. See also http://www.mts.jhu.edu/~fill/ and
http://www.mathcs.carleton.edu/faculty/bdobrow/. Submitted for publication in
May, 200
Processes with Long Memory: Regenerative Construction and Perfect Simulation
We present a perfect simulation algorithm for stationary processes indexed by
Z, with summable memory decay. Depending on the decay, we construct the process
on finite or semi-infinite intervals, explicitly from an i.i.d. uniform
sequence. Even though the process has infinite memory, its value at time 0
depends only on a finite, but random, number of these uniform variables. The
algorithm is based on a recent regenerative construction of these measures by
Ferrari, Maass, Mart{\'\i}nez and Ney. As applications, we discuss the perfect
simulation of binary autoregressions and Markov chains on the unit interval.Comment: 27 pages, one figure. Version accepted by Annals of Applied
Probability. Small changes with respect to version
Probabilistic cellular automata, invariant measures, and perfect sampling
A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The
cells are updated synchronously and independently, according to a distribution
depending on a finite neighborhood. We investigate the ergodicity of this
Markov chain. A classical cellular automaton is a particular case of PCA. For a
1-dimensional cellular automaton, we prove that ergodicity is equivalent to
nilpotency, and is therefore undecidable. We then propose an efficient perfect
sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm
does not assume any monotonicity property of the local rule. It is based on a
bounding process which is shown to be also a PCA. Last, we focus on the PCA
Majority, whose asymptotic behavior is unknown, and perform numerical
experiments using the perfect sampling procedure
Multiprocess parallel antithetic coupling for backward and forward Markov Chain Monte Carlo
Antithetic coupling is a general stratification strategy for reducing Monte
Carlo variance without increasing the simulation size. The use of the
antithetic principle in the Monte Carlo literature typically employs two strata
via antithetic quantile coupling. We demonstrate here that further
stratification, obtained by using k>2 (e.g., k=3-10) antithetically coupled
variates, can offer substantial additional gain in Monte Carlo efficiency, in
terms of both variance and bias. The reason for reduced bias is that
antithetically coupled chains can provide a more dispersed search of the state
space than multiple independent chains. The emerging area of perfect simulation
provides a perfect setting for implementing the k-process parallel antithetic
coupling for MCMC because, without antithetic coupling, this class of methods
delivers genuine independent draws. Furthermore, antithetic backward coupling
provides a very convenient theoretical tool for investigating antithetic
forward coupling. However, the generation of k>2 antithetic variates that are
negatively associated, that is, they preserve negative correlation under
monotone transformations, and extremely antithetic, that is, they are as
negatively correlated as possible, is more complicated compared to the case
with k=2. In this paper, we establish a theoretical framework for investigating
such issues. Among the generating methods that we compare, Latin hypercube
sampling and its iterative extension appear to be general-purpose choices,
making another direct link between Monte Carlo and quasi Monte Carlo.Comment: Published at http://dx.doi.org/10.1214/009053604000001075 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Note on rejection sampling and exact sampling with the Metropolised independence sampler
This short note shows a close relationship between standard rejection sampling and exact sampling by coupling from the past applied to a Metropolised independence sampler. I now know that this idea, first presented as a ten-minute tea-time talk, is probably a duplicate of an unavailable work (Cai 1997), and is closely related to a paper by Jun S. Liu (1996), who provides a much more detailed analysis. Perhaps this exposition will be of interest to some readers
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