41 research outputs found
Efficient Gibbs Sampling for Markov Switching GARCH Models
We develop efficient simulation techniques for Bayesian inference on
switching GARCH models. Our contribution to existing literature is manifold.
First, we discuss different multi-move sampling techniques for Markov Switching
(MS) state space models with particular attention to MS-GARCH models. Our
multi-move sampling strategy is based on the Forward Filtering Backward
Sampling (FFBS) applied to an approximation of MS-GARCH. Another important
contribution is the use of multi-point samplers, such as the Multiple-Try
Metropolis (MTM) and the Multiple trial Metropolize Independent Sampler, in
combination with FFBS for the MS-GARCH process. In this sense we ex- tend to
the MS state space models the work of So [2006] on efficient MTM sampler for
continuous state space models. Finally, we suggest to further improve the
sampler efficiency by introducing the antithetic sampling of Craiu and Meng
[2005] and Craiu and Lemieux [2007] within the FFBS. Our simulation experiments
on MS-GARCH model show that our multi-point and multi-move strategies allow the
sampler to gain efficiency when compared with single-move Gibbs sampling.Comment: 38 pages, 7 figure
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
Construction of weakly CUD sequences for MCMC sampling
In Markov chain Monte Carlo (MCMC) sampling considerable thought goes into
constructing random transitions. But those transitions are almost always driven
by a simulated IID sequence. Recently it has been shown that replacing an IID
sequence by a weakly completely uniformly distributed (WCUD) sequence leads to
consistent estimation in finite state spaces. Unfortunately, few WCUD sequences
are known. This paper gives general methods for proving that a sequence is
WCUD, shows that some specific sequences are WCUD, and shows that certain
operations on WCUD sequences yield new WCUD sequences. A numerical example on a
42 dimensional continuous Gibbs sampler found that some WCUD inputs sequences
produced variance reductions ranging from tens to hundreds for posterior means
of the parameters, compared to IID inputs.Comment: Published in at http://dx.doi.org/10.1214/07-EJS162 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
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
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