37,860 research outputs found
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
Highly efficient Bayesian joint inversion for receiver-based data and its application to lithospheric structure beneath the southern Korean Peninsula
With the deployment of extensive seismic arrays, systematic and efficient parameter and uncertainty estimation is of increasing importance and can provide reliable, regional models for crustal and upper-mantle structure.We present an efficient Bayesian method for the joint inversion of surface-wave dispersion and receiver-function data that combines trans-dimensional (trans-D) model selection in an optimization phase with subsequent rigorous parameter uncertainty estimation. Parameter and uncertainty estimation depend strongly on the chosen parametrization such that meaningful regional comparison requires quantitative model selection that can be carried out efficiently at several sites. While significant progress has been made for model selection (e.g. trans-D inference) at individual sites, the lack of efficiency can prohibit application to large data volumes or cause questionable results due to lack of convergence. Studies that address large numbers of data sets have mostly ignored model selection in favour of more efficient/simple estimation techniques (i.e. focusing on uncertainty estimation but employing ad-hoc model choices). Our approach consists of a two-phase inversion that combines trans-D optimization to select the most probable parametrization with subsequent Bayesian sampling for uncertainty estimation given that parametrization. The trans-D optimization is implemented here by replacing the likelihood function with the Bayesian information criterion (BIC). The BIC provides constraints on model complexity that facilitate the search for an optimal parametrization. Parallel tempering (PT) is applied as an optimization algorithm. After optimization, the optimal model choice is identified by the minimum BIC value from all PT chains. Uncertainty estimation is then carried out in fixed dimension. Data errors are estimated as part of the inference problem by a combination of empirical and hierarchical estimation. Data covariance matrices are estimated from data residuals (the difference between prediction and observation) and periodically updated. In addition, a scaling factor for the covariance matrix magnitude is estimated as part of the inversion. The inversion is applied to both simulated and observed data that consist of phase- and group-velocity dispersion curves (Rayleigh wave), and receiver functions. The simulation results show that model complexity and important features are well estimated by the fixed dimensional posterior probability density. Observed data for stations in different tectonic regions of the southern Korean Peninsula are considered. The results are consistent with published results, but important features are better constrained than in previous regularized inversions and are more consistent across the stations. For example, resolution of crustal and Moho interfaces, and absolute values and gradients of velocities in lower crust and upper mantle are better constrained
Bayesian Modelling and Inference on Mixtures of Distributions.
bayesian models;
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Martingale proofs of many-server heavy-traffic limits for Markovian queues
This is an expository review paper illustrating the ``martingale method'' for
proving many-server heavy-traffic stochastic-process limits for queueing
models, supporting diffusion-process approximations. Careful treatment is given
to an elementary model -- the classical infinite-server model , but
models with finitely many servers and customer abandonment are also treated.
The Markovian stochastic process representing the number of customers in the
system is constructed in terms of rate-1 Poisson processes in two ways: (i)
through random time changes and (ii) through random thinnings. Associated
martingale representations are obtained for these constructions by applying,
respectively: (i) optional stopping theorems where the random time changes are
the stopping times and (ii) the integration theorem associated with random
thinning of a counting process. Convergence to the diffusion process limit for
the appropriate sequence of scaled queueing processes is obtained by applying
the continuous mapping theorem. A key FCLT and a key FWLLN in this framework
are established both with and without applying martingales.Comment: Published in at http://dx.doi.org/10.1214/06-PS091 the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Spontaneous magnetisation in the plane
The Arak process is a solvable stochastic process which generates coloured
patterns in the plane. Patterns are made up of a variable number of random
non-intersecting polygons. We show that the distribution of Arak process states
is the Gibbs distribution of its states in thermodynamic equilibrium in the
grand canonical ensemble. The sequence of Gibbs distributions form a new model
parameterised by temperature. We prove that there is a phase transition in this
model, for some non-zero temperature. We illustrate this conclusion with
simulation results. We measure the critical exponents of this off-lattice model
and find they are consistent with those of the Ising model in two dimensions.Comment: 23 pages numbered -1,0...21, 8 figure
SimInf: An R package for Data-driven Stochastic Disease Spread Simulations
We present the R package SimInf which provides an efficient and very flexible
framework to conduct data-driven epidemiological modeling in realistic large
scale disease spread simulations. The framework integrates infection dynamics
in subpopulations as continuous-time Markov chains using the Gillespie
stochastic simulation algorithm and incorporates available data such as births,
deaths and movements as scheduled events at predefined time-points. Using C
code for the numerical solvers and OpenMP to divide work over multiple
processors ensures high performance when simulating a sample outcome. One of
our design goal was to make SimInf extendable and enable usage of the numerical
solvers from other R extension packages in order to facilitate complex
epidemiological research. In this paper, we provide a technical description of
the framework and demonstrate its use on some basic examples. We also discuss
how to specify and extend the framework with user-defined models.Comment: The manual has been updated to the latest version of SimInf (v6.0.0).
41 pages, 16 figure
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
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