2,164 research outputs found
On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions
The Metropolis algorithm is arguably the most fundamental Markov chain Monte
Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the
desired distribution in the case of multivariate binary distributions (e.g.,
Ising models or stochastic neural networks such as Boltzmann machines) if the
variables (sites or neurons) are updated in a fixed order, a setting commonly
used in practice. The reason is that the corresponding Markov chain may not be
irreducible. We propose a modified Metropolis transition operator that behaves
almost always identically to the standard Metropolis operator and prove that it
ensures irreducibility and convergence to the limiting distribution in the
multivariate binary case with fixed-order updates. The result provides an
explanation for the behaviour of Metropolis MCMC in that setting and closes a
long-standing theoretical gap. We experimentally studied the standard and
modified Metropolis operator for models were they actually behave differently.
If the standard algorithm also converges, the modified operator exhibits
similar (if not better) performance in terms of convergence speed
Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process
We present an adaptive method for the automatic scaling of Random-Walk
Metropolis-Hastings algorithms, which quickly and robustly identifies the
scaling factor that yields a specified overall sampler acceptance probability.
Our method relies on the use of the Robbins-Monro search process, whose
performance is determined by an unknown steplength constant. We give a very
simple estimator of this constant for proposal distributions that are
univariate or multivariate normal, together with a sampling algorithm for
automating the method. The effectiveness of the algorithm is demonstrated with
both simulated and real data examples. This approach could be implemented as a
useful component in more complex adaptive Markov chain Monte Carlo algorithms,
or as part of automated software packages
Markov Chain Monte Carlo Based on Deterministic Transformations
In this article we propose a novel MCMC method based on deterministic
transformations T: X x D --> X where X is the state-space and D is some set
which may or may not be a subset of X. We refer to our new methodology as
Transformation-based Markov chain Monte Carlo (TMCMC). One of the remarkable
advantages of our proposal is that even if the underlying target distribution
is very high-dimensional, deterministic transformation of a one-dimensional
random variable is sufficient to generate an appropriate Markov chain that is
guaranteed to converge to the high-dimensional target distribution. Apart from
clearly leading to massive computational savings, this idea of
deterministically transforming a single random variable very generally leads to
excellent acceptance rates, even though all the random variables associated
with the high-dimensional target distribution are updated in a single block.
Since it is well-known that joint updating of many random variables using
Metropolis-Hastings (MH) algorithm generally leads to poor acceptance rates,
TMCMC, in this regard, seems to provide a significant advance. We validate our
proposal theoretically, establishing the convergence properties. Furthermore,
we show that TMCMC can be very effectively adopted for simulating from doubly
intractable distributions.
TMCMC is compared with MH using the well-known Challenger data, demonstrating
the effectiveness of of the former in the case of highly correlated variables.
Moreover, we apply our methodology to a challenging posterior simulation
problem associated with the geostatistical model of Diggle et al. (1998),
updating 160 unknown parameters jointly, using a deterministic transformation
of a one-dimensional random variable. Remarkable computational savings as well
as good convergence properties and acceptance rates are the results.Comment: 28 pages, 3 figures; Longer abstract inside articl
A Bayesian approach to the study of white dwarf binaries in LISA data: The application of a reversible jump Markov chain Monte Carlo method
The Laser Interferometer Space Antenna (LISA) defines new demands on data
analysis efforts in its all-sky gravitational wave survey, recording
simultaneously thousands of galactic compact object binary foreground sources
and tens to hundreds of background sources like binary black hole mergers and
extreme mass ratio inspirals. We approach this problem with an adaptive and
fully automatic Reversible Jump Markov Chain Monte Carlo sampler, able to
sample from the joint posterior density function (as established by Bayes
theorem) for a given mixture of signals "out of the box'', handling the total
number of signals as an additional unknown parameter beside the unknown
parameters of each individual source and the noise floor. We show in examples
from the LISA Mock Data Challenge implementing the full response of LISA in its
TDI description that this sampler is able to extract monochromatic Double White
Dwarf signals out of colored instrumental noise and additional foreground and
background noise successfully in a global fitting approach. We introduce 2
examples with fixed number of signals (MCMC sampling), and 1 example with
unknown number of signals (RJ-MCMC), the latter further promoting the idea
behind an experimental adaptation of the model indicator proposal densities in
the main sampling stage. We note that the experienced runtimes and degeneracies
in parameter extraction limit the shown examples to the extraction of a low but
realistic number of signals.Comment: 18 pages, 9 figures, 3 tables, accepted for publication in PRD,
revised versio
Sequential Monte Carlo EM for multivariate probit models
Multivariate probit models (MPM) have the appealing feature of capturing some
of the dependence structure between the components of multidimensional binary
responses. The key for the dependence modelling is the covariance matrix of an
underlying latent multivariate Gaussian. Most approaches to MLE in multivariate
probit regression rely on MCEM algorithms to avoid computationally intensive
evaluations of multivariate normal orthant probabilities. As an alternative to
the much used Gibbs sampler a new SMC sampler for truncated multivariate
normals is proposed. The algorithm proceeds in two stages where samples are
first drawn from truncated multivariate Student distributions and then
further evolved towards a Gaussian. The sampler is then embedded in a MCEM
algorithm. The sequential nature of SMC methods can be exploited to design a
fully sequential version of the EM, where the samples are simply updated from
one iteration to the next rather than resampled from scratch. Recycling the
samples in this manner significantly reduces the computational cost. An
alternative view of the standard conditional maximisation step provides the
basis for an iterative procedure to fully perform the maximisation needed in
the EM algorithm. The identifiability of MPM is also thoroughly discussed. In
particular, the likelihood invariance can be embedded in the EM algorithm to
ensure that constrained and unconstrained maximisation are equivalent. A simple
iterative procedure is then derived for either maximisation which takes
effectively no computational time. The method is validated by applying it to
the widely analysed Six Cities dataset and on a higher dimensional simulated
example. Previous approaches to the Six Cities overly restrict the parameter
space but, by considering the correct invariance, the maximum likelihood is
quite naturally improved when treating the full unrestricted model.Comment: 26 pages, 2 figures. In press, Computational Statistics & Data
Analysi
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