1,788 research outputs found
N-fold way simulated tempering for pairwise interaction point processes
Pairwise interaction point processes with strong interaction are usually difficult to
sample. We discuss how Besag lattice processes can be used in a simulated tempering
MCMC scheme to help with the simulation of such processes. We show how
the N-fold way algorithm can be used to sample the lattice processes efficiently
and introduce the N-fold way algorithm into our simulated tempering scheme. To
calibrate the simulated tempering scheme we use the Wang-Landau algorithm
Scalable iterative methods for sampling from massive Gaussian random vectors
Sampling from Gaussian Markov random fields (GMRFs), that is multivariate
Gaussian ran- dom vectors that are parameterised by the inverse of their
covariance matrix, is a fundamental problem in computational statistics. In
this paper, we show how we can exploit arbitrarily accu- rate approximations to
a GMRF to speed up Krylov subspace sampling methods. We also show that these
methods can be used when computing the normalising constant of a large
multivariate Gaussian distribution, which is needed for both any
likelihood-based inference method. The method we derive is also applicable to
other structured Gaussian random vectors and, in particu- lar, we show that
when the precision matrix is a perturbation of a (block) circulant matrix, it
is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression
In this work we perform a meta-analysis of neuroimaging data, consisting of
locations of peak activations identified in 162 separate studies on emotion.
Neuroimaging meta-analyses are typically performed using kernel-based methods.
However, these methods require the width of the kernel to be set a priori and
to be constant across the brain. To address these issues, we propose a fully
Bayesian nonparametric binary regression method to perform neuroimaging
meta-analyses. In our method, each location (or voxel) has a probability of
being a peak activation, and the corresponding probability function is based on
a spatially adaptive Gaussian Markov random field (GMRF). We also include
parameters in the model to robustify the procedure against miscoding of the
voxel response. Posterior inference is implemented using efficient MCMC
algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1
(2006) 145--168]. Our method allows the probability function to be locally
adaptive with respect to the covariates, that is, to be smooth in one region of
the covariate space and wiggly or even discontinuous in another. Posterior
miscoding probabilities for each of the identified voxels can also be obtained,
identifying voxels that may have been falsely classified as being activated.
Simulation studies and application to the emotion neuroimaging data indicate
that our method is superior to standard kernel-based methods.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS523 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Construction of weakly CUD sequences for MCMC sampling
In Markov chain Monte Carlo (MCMC) sampling considerable thought goes into
constructing random transitions. But those transitions are almost always driven
by a simulated IID sequence. Recently it has been shown that replacing an IID
sequence by a weakly completely uniformly distributed (WCUD) sequence leads to
consistent estimation in finite state spaces. Unfortunately, few WCUD sequences
are known. This paper gives general methods for proving that a sequence is
WCUD, shows that some specific sequences are WCUD, and shows that certain
operations on WCUD sequences yield new WCUD sequences. A numerical example on a
42 dimensional continuous Gibbs sampler found that some WCUD inputs sequences
produced variance reductions ranging from tens to hundreds for posterior means
of the parameters, compared to IID inputs.Comment: Published in at http://dx.doi.org/10.1214/07-EJS162 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
Kinetic energy choice in Hamiltonian/hybrid Monte Carlo
We consider how different choices of kinetic energy in Hamiltonian Monte
Carlo affect algorithm performance. To this end, we introduce two quantities
which can be easily evaluated, the composite gradient and the implicit noise.
Results are established on integrator stability and geometric convergence, and
we show that choices of kinetic energy that result in heavy-tailed momentum
distributions can exhibit an undesirable negligible moves property, which we
define. A general efficiency-robustness trade off is outlined, and
implementations which rely on approximate gradients are also discussed. Two
numerical studies illustrate our theoretical findings, showing that the
standard choice which results in a Gaussian momentum distribution is not always
optimal in terms of either robustness or efficiency.Comment: 15 pages (+7 page supplement, included here as an appendix), 2
figures (+1 in supplement
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