327 research outputs found
Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains
A -irreducible and aperiodic Markov chain with stationary probability
distribution will converge to its stationary distribution from almost all
starting points. The property of Harris recurrence allows us to replace
``almost all'' by ``all,'' which is potentially important when running Markov
chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms
are known to be Harris recurrent. In this paper, we consider conditions under
which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are
not Harris recurrent. We present a simple but natural two-dimensional
counter-example showing how Harris recurrence can fail, and also a variety of
positive results which guarantee Harris recurrence. We also present some open
problems. We close with a discussion of the practical implications for MCMC
algorithms.Comment: Published at http://dx.doi.org/10.1214/105051606000000510 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Accelerating Parallel Tempering: Quantile Tempering Algorithm (QuanTA)
Using MCMC to sample from a target distribution, on a
-dimensional state space can be a difficult and computationally expensive
problem. Particularly when the target exhibits multimodality, then the
traditional methods can fail to explore the entire state space and this results
in a bias sample output. Methods to overcome this issue include the parallel
tempering algorithm which utilises an augmented state space approach to help
the Markov chain traverse regions of low probability density and reach other
modes. This method suffers from the curse of dimensionality which dramatically
slows the transfer of mixing information from the auxiliary targets to the
target of interest as . This paper introduces a novel
prototype algorithm, QuanTA, that uses a Gaussian motivated transformation in
an attempt to accelerate the mixing through the temperature schedule of a
parallel tempering algorithm. This new algorithm is accompanied by a
comprehensive theoretical analysis quantifying the improved efficiency and
scalability of the approach; concluding that under weak regularity conditions
the new approach gives accelerated mixing through the temperature schedule.
Empirical evidence of the effectiveness of this new algorithm is illustrated on
canonical examples
Minimising MCMC variance via diffusion limits, with an application to simulated tempering
We derive new results comparing the asymptotic variance of diffusions by
writing them as appropriate limits of discrete-time birth-death chains which
themselves satisfy Peskun orderings. We then apply our results to simulated
tempering algorithms to establish which choice of inverse temperatures
minimises the asymptotic variance of all functionals and thus leads to the most
efficient MCMC algorithm.Comment: Published in at http://dx.doi.org/10.1214/12-AAP918 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Likelihood-based inference for correlated diffusions
We address the problem of likelihood based inference for correlated diffusion
processes using Markov chain Monte Carlo (MCMC) techniques. Such a task
presents two interesting problems. First, the construction of the MCMC scheme
should ensure that the correlation coefficients are updated subject to the
positive definite constraints of the diffusion matrix. Second, a diffusion may
only be observed at a finite set of points and the marginal likelihood for the
parameters based on these observations is generally not available. We overcome
the first issue by using the Cholesky factorisation on the diffusion matrix. To
deal with the likelihood unavailability, we generalise the data augmentation
framework of Roberts and Stramer (2001 Biometrika 88(3):603-621) to
d-dimensional correlated diffusions including multivariate stochastic
volatility models. Our methodology is illustrated through simulation based
experiments and with daily EUR /USD, GBP/USD rates together with their implied
volatilities
Adaptive Gibbs samplers and related MCMC methods
We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs
samplers, which update their selection probabilities (and perhaps also their
proposal distributions) on the fly during a run by learning as they go in an
attempt to optimize the algorithm. We present a cautionary example of how even
a simple-seeming adaptive Gibbs sampler may fail to converge. We then present
various positive results guaranteeing convergence of adaptive Gibbs samplers
under certain conditions.Comment: Published in at http://dx.doi.org/10.1214/11-AAP806 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note:
substantial text overlap with arXiv:1001.279
An approximation scheme for quasi-stationary distributions of killed diffusions
In this paper we study the asymptotic behavior of the normalized weighted
empirical occupation measures of a diffusion process on a compact manifold
which is killed at a smooth rate and then regenerated at a random location,
distributed according to the weighted empirical occupation measure. We show
that the weighted occupation measures almost surely comprise an asymptotic
pseudo-trajectory for a certain deterministic measure-valued semiflow, after
suitably rescaling the time, and that with probability one they converge to the
quasi-stationary distribution of the killed diffusion. These results provide
theoretical justification for a scalable quasi-stationary Monte Carlo method
for sampling from Bayesian posterior distributions.Comment: v2: revised version, 29 pages, 1 figur
Inference for stochastic volatility model using time change transformations
We address the problem of parameter estimation for diffusion driven stochastic volatility models through Markov chain Monte Carlo (MCMC). To avoid degeneracy issues we introduce an innovative reparametrisation defined through transformations that operate on the time scale of the diffusion. A novel MCMC scheme which overcomes the inherent difficulties of time change transformations is also presented. The algorithm is fast to implement and applies to models with stochastic volatility. The methodology is tested through simulation based experiments and illustrated on data consisting of US treasury bill rates.Imputation, Markov chain Monte Carlo, Stochastic volatility
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