6,075 research outputs found
Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with
Exploration of the intractable posterior distributions associated with
Bayesian versions of the general linear mixed model is often performed using
Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is
used, then there is a simple two-block Gibbs sampler available. Rom\'{a}n and
Hobert [Linear Algebra Appl. 473 (2015) 54-77] showed that, when the priors are
proper and the matrix has full column rank, the Markov chains underlying
these Gibbs samplers are nearly always geometrically ergodic. In this paper,
Rom\'{a}n and Hobert's (2015) result is extended by allowing improper priors on
the variance components, and, more importantly, by removing all assumptions on
the matrix. So, not only is allowed to be (column) rank deficient,
which provides additional flexibility in parameterizing the fixed effects, it
is also allowed to have more columns than rows, which is necessary in the
increasingly important situation where . The full rank assumption on
is at the heart of Rom\'{a}n and Hobert's (2015) proof. Consequently, the
extension to unrestricted requires a substantially different analysis.Comment: Published at http://dx.doi.org/10.3150/15-BEJ749 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
Information-geometric Markov Chain Monte Carlo methods using Diffusions
Recent work incorporating geometric ideas in Markov chain Monte Carlo is
reviewed in order to highlight these advances and their possible application in
a range of domains beyond Statistics. A full exposition of Markov chains and
their use in Monte Carlo simulation for Statistical inference and molecular
dynamics is provided, with particular emphasis on methods based on Langevin
diffusions. After this geometric concepts in Markov chain Monte Carlo are
introduced. A full derivation of the Langevin diffusion on a Riemannian
manifold is given, together with a discussion of appropriate Riemannian metric
choice for different problems. A survey of applications is provided, and some
open questions are discussed.Comment: 22 pages, 2 figure
Curvature and Concentration of Hamiltonian Monte Carlo in High Dimensions
In this article, we analyze Hamiltonian Monte Carlo (HMC) by placing it in
the setting of Riemannian geometry using the Jacobi metric, so that each step
corresponds to a geodesic on a suitable Riemannian manifold. We then combine
the notion of curvature of a Markov chain due to Joulin and Ollivier with the
classical sectional curvature from Riemannian geometry to derive error bounds
for HMC in important cases, where we have positive curvature. These cases
include several classical distributions such as multivariate Gaussians, and
also distributions arising in the study of Bayesian image registration. The
theoretical development suggests the sectional curvature as a new diagnostic
tool for convergence for certain Markov chains.Comment: Comments welcom
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