9,621 research outputs found
Sampling constrained probability distributions using Spherical Augmentation
Statistical models with constrained probability distributions are abundant in
machine learning. Some examples include regression models with norm constraints
(e.g., Lasso), probit, many copula models, and latent Dirichlet allocation
(LDA). Bayesian inference involving probability distributions confined to
constrained domains could be quite challenging for commonly used sampling
algorithms. In this paper, we propose a novel augmentation technique that
handles a wide range of constraints by mapping the constrained domain to a
sphere in the augmented space. By moving freely on the surface of this sphere,
sampling algorithms handle constraints implicitly and generate proposals that
remain within boundaries when mapped back to the original space. Our proposed
method, called {Spherical Augmentation}, provides a mathematically natural and
computationally efficient framework for sampling from constrained probability
distributions. We show the advantages of our method over state-of-the-art
sampling algorithms, such as exact Hamiltonian Monte Carlo, using several
examples including truncated Gaussian distributions, Bayesian Lasso, Bayesian
bridge regression, reconstruction of quantized stationary Gaussian process, and
LDA for topic modeling.Comment: 41 pages, 13 figure
Dynamic density estimation with diffusive Dirichlet mixtures
We introduce a new class of nonparametric prior distributions on the space of
continuously varying densities, induced by Dirichlet process mixtures which
diffuse in time. These select time-indexed random functions without jumps,
whose sections are continuous or discrete distributions depending on the choice
of kernel. The construction exploits the widely used stick-breaking
representation of the Dirichlet process and induces the time dependence by
replacing the stick-breaking components with one-dimensional Wright-Fisher
diffusions. These features combine appealing properties of the model, inherited
from the Wright-Fisher diffusions and the Dirichlet mixture structure, with
great flexibility and tractability for posterior computation. The construction
can be easily extended to multi-parameter GEM marginal states, which include,
for example, the Pitman--Yor process. A full inferential strategy is detailed
and illustrated on simulated and real data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ681 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Revealing networks from dynamics: an introduction
What can we learn from the collective dynamics of a complex network about its
interaction topology? Taking the perspective from nonlinear dynamics, we
briefly review recent progress on how to infer structural connectivity (direct
interactions) from accessing the dynamics of the units. Potential applications
range from interaction networks in physics, to chemical and metabolic
reactions, protein and gene regulatory networks as well as neural circuits in
biology and electric power grids or wireless sensor networks in engineering.
Moreover, we briefly mention some standard ways of inferring effective or
functional connectivity.Comment: Topical review, 48 pages, 7 figure
Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields
A numerical integration method for guiding-center orbits of charged particles
in toroidal fusion devices with three-dimensional field geometry is described.
Here, high order interpolation of electromagnetic fields in space is replaced
by a special linear interpolation, leading to locally linear Hamiltonian
equations of motion with piecewise constant coefficients. This approach reduces
computational effort and noise sensitivity while the conservation of total
energy, magnetic moment and phase space volume is retained. The underlying
formulation treats motion in piecewise linear fields exactly and thus preserves
the non-canonical symplectic form. The algorithm itself is only quasi-geometric
due to a series expansion in the orbit parameter. For practical purposes an
expansion to the fourth order retains geometric properties down to computer
accuracy in typical examples. When applied to collisionless guiding-center
orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator
configuration, the method demonstrates stable long-term orbit dynamics
conserving invariants. In Monte Carlo evaluation of transport coefficients, the
computational efficiency of quasi-geometric integration is an order of
magnitude higher than with a standard fourth order Runge-Kutta integrator.Comment: 38 pages, 11 figure
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