4,098 research outputs found
Metropolis Sampling
Monte Carlo (MC) sampling methods are widely applied in Bayesian inference,
system simulation and optimization problems. The Markov Chain Monte Carlo
(MCMC) algorithms are a well-known class of MC methods which generate a Markov
chain with the desired invariant distribution. In this document, we focus on
the Metropolis-Hastings (MH) sampler, which can be considered as the atom of
the MCMC techniques, introducing the basic notions and different properties. We
describe in details all the elements involved in the MH algorithm and the most
relevant variants. Several improvements and recent extensions proposed in the
literature are also briefly discussed, providing a quick but exhaustive
overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201
A multiple-try Metropolis-Hastings algorithm with tailored proposals
We present a new multiple-try Metropolis-Hastings algorithm designed to be
especially beneficial when a tailored proposal distribution is available. The
algorithm is based on a given acyclic graph , where one of the nodes in ,
say, contains the current state of the Markov chain and the remaining nodes
contain proposed states generated by applying the tailored proposal
distribution. The Metropolis-Hastings algorithm alternates between two types of
updates. The first update type is using the tailored proposal distribution to
generate new states in all nodes in except in node . The second update
type is generating a new value for , thereby changing the value of the
current state. We evaluate the effectiveness of the proposed scheme in an
example with previously defined target and proposal distributions
Interacting Multiple Try Algorithms with Different Proposal Distributions
We propose a new class of interacting Markov chain Monte Carlo (MCMC)
algorithms designed for increasing the efficiency of a modified multiple-try
Metropolis (MTM) algorithm. The extension with respect to the existing MCMC
literature is twofold. The sampler proposed extends the basic MTM algorithm by
allowing different proposal distributions in the multiple-try generation step.
We exploit the structure of the MTM algorithm with different proposal
distributions to naturally introduce an interacting MTM mechanism (IMTM) that
expands the class of population Monte Carlo methods. We show the validity of
the algorithm and discuss the choice of the selection weights and of the
different proposals. We provide numerical studies which show that the new
algorithm can perform better than the basic MTM algorithm and that the
interaction mechanism allows the IMTM to efficiently explore the state space
On the flexibility of the design of Multiple Try Metropolis schemes
The Multiple Try Metropolis (MTM) method is a generalization of the classical
Metropolis-Hastings algorithm in which the next state of the chain is chosen
among a set of samples, according to normalized weights. In the literature,
several extensions have been proposed. In this work, we show and remark upon
the flexibility of the design of MTM-type methods, fulfilling the detailed
balance condition. We discuss several possibilities and show different
numerical results
Order of magnitude time-reversible Markov chains and characterization of clustering processes
We introduce the notion of order of magnitude reversibility
(OM-reversibility) in Markov chains that are parametrized by a positive
parameter \ep. OM-reversibility is a weaker condition than reversibility, and
requires only the knowledge of order of magnitude of the transition
probabilities. For an irreducible, OM-reversible Markov chain on a finite state
space, we prove that the stationary distribution satisfies order of magnitude
detailed balance (analog of detailed balance in reversible Markov chains). The
result characterizes the states with positive probability in the limit of the
stationary distribution as \ep \to 0, which finds an important application in
the case of singularly perturbed Markov chains that are reducible for \ep=0.
We show that OM-reversibility occurs naturally in macroscopic systems,
involving many interacting particles. Clustering is a common phenomenon in
biological systems, in which particles or molecules aggregate at one location.
We give a simple condition on the transition probabilities in an interacting
particle Markov chain that characterizes clustering. We show that such
clustering processes are OM-reversible, and we find explicitly the order of
magnitude of the stationary distribution. Further, we show that the single pole
states, in which all particles are at a single vertex, are the only states with
positive probability in the limit of the stationary distribution as the rate of
diffusion goes to zero.Comment: 22 pages, 3 figure
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