4,436 research outputs found
Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals
Estimation of parameters of a diffusion based on discrete time observations
poses a difficult problem due to the lack of a closed form expression for the
likelihood. From a Bayesian computational perspective it can be casted as a
missing data problem where the diffusion bridges in between discrete-time
observations are missing. The computational problem can then be dealt with
using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown
parameters appear in the diffusion coefficient, direct implementation of
data-augmentation results in a Markov chain that is reducible. Furthermore,
data-augmentation requires efficient sampling of diffusion bridges, which can
be difficult, especially in the multidimensional case.
We present a general framework to deal with with these problems that does not
rely on discretisation. The construction generalises previous approaches and
sheds light on the assumptions necessary to make these approaches work. We
define a random-walk type Metropolis-Hastings sampler for updating diffusion
bridges. Our methods are illustrated using guided proposals for sampling
diffusion bridges. These are Markov processes obtained by adding a guiding term
to the drift of the diffusion. We give general guidelines on the construction
of these proposals and introduce a time change and scaling of the guided
proposal that reduces discretisation error. Numerical examples demonstrate the
performance of our methods
Stochastic Methods for Zero Energy Quantum Scattering
We investigate the use of stochastic methods for zero energy quantum
scattering based on a path integral approach. With the application to the
scattering of a projectile from a nuclear many body target in mind, we use the
potential scattering of a particle as a test for the accuracy and efficiency of
several methods. To be able to deal with complex potentials, we introduce a
path sampling action and a modified scattering observable. The approaches
considered are the random walk, where the points of a path are sequentially
generated, and the Langevin algorithm, which updates an entire path. Several
improvements are investigated. A cluster algorithm for dealing with scattering
problems is finally proposed, which shows the best accuracy and stability.Comment: 40 pages LaTeX, 1 Postscript file containig 20 figures; execute
main.tex file, which automatically will include other file
Hamiltonian Monte Carlo Without Detailed Balance
We present a method for performing Hamiltonian Monte Carlo that largely
eliminates sample rejection for typical hyperparameters. In situations that
would normally lead to rejection, instead a longer trajectory is computed until
a new state is reached that can be accepted. This is achieved using Markov
chain transitions that satisfy the fixed point equation, but do not satisfy
detailed balance. The resulting algorithm significantly suppresses the random
walk behavior and wasted function evaluations that are typically the
consequence of update rejection. We demonstrate a greater than factor of two
improvement in mixing time on three test problems. We release the source code
as Python and MATLAB packages.Comment: Accepted conference submission to ICML 2014 and also featured in a
special edition of JMLR. Since updated to include additional literature
citation
Tuning the generalized Hybrid Monte Carlo algorithm
We discuss the analytic computation of autocorrelation functions for the
generalized Hybrid Monte Carlo algorithm applied to free field theory and
compare the results with numerical results for the spin model in two
dimensions. We explain how the dynamical critical exponent for some
operators may be reduced from two to one by tuning the amount of randomness
introduced by the updating procedure, and why critical slowing down is not a
problem for other operators.Comment: 4 pages, to be published in the Proceedings of Lattice 95, uuencoded
PostScript fil
Wormhole Hamiltonian Monte Carlo
In machine learning and statistics, probabilistic inference involving
multimodal distributions is quite difficult. This is especially true in high
dimensional problems, where most existing algorithms cannot easily move from
one mode to another. To address this issue, we propose a novel Bayesian
inference approach based on Markov Chain Monte Carlo. Our method can
effectively sample from multimodal distributions, especially when the dimension
is high and the modes are isolated. To this end, it exploits and modifies the
Riemannian geometric properties of the target distribution to create
\emph{wormholes} connecting modes in order to facilitate moving between them.
Further, our proposed method uses the regeneration technique in order to adapt
the algorithm by identifying new modes and updating the network of wormholes
without affecting the stationary distribution. To find new modes, as opposed to
rediscovering those previously identified, we employ a novel mode searching
algorithm that explores a \emph{residual energy} function obtained by
subtracting an approximate Gaussian mixture density (based on previously
discovered modes) from the target density function
A New Technique for Sampling Multi-Modal Distributions
In this paper we demonstrate that multi-modal Probability Distribution
Functions (PDFs) may be efficiently sampled using an algorithm originally
developed for numerical integrations by Monte-Carlo methods. This algorithm can
be used to generate an input PDF which can be used as an independence sampler
in a Metropolis-Hastings chain to sample otherwise troublesome
distributions.Some examples in one two and five dimensions are worked out.Comment: One ps figure; submitted to "Journal of Computational Physics
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