3,387 research outputs found
A fast Metropolis-Hastings method for generating random correlation matrices
We propose a novel Metropolis-Hastings algorithm to sample uniformly from the
space of correlation matrices. Existing methods in the literature are based on
elaborated representations of a correlation matrix, or on complex
parametrizations of it. By contrast, our method is intuitive and simple, based
the classical Cholesky factorization of a positive definite matrix and Markov
chain Monte Carlo theory. We perform a detailed convergence analysis of the
resulting Markov chain, and show how it benefits from fast convergence, both
theoretically and empirically. Furthermore, in numerical experiments our
algorithm is shown to be significantly faster than the current alternative
approaches, thanks to its simple yet principled approach.Comment: 8 pages, 3 figures, 2018 conferenc
Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution
Direct simulation of biomolecular dynamics in thermal equilibrium is
challenging due to the metastable nature of conformation dynamics and the
computational cost of molecular dynamics. Biased or enhanced sampling methods
may improve the convergence of expectation values of equilibrium probabilities
and expectation values of stationary quantities significantly. Unfortunately
the convergence of dynamic observables such as correlation functions or
timescales of conformational transitions relies on direct equilibrium
simulations. Markov state models are well suited to describe both, stationary
properties and properties of slow dynamical processes of a molecular system, in
terms of a transition matrix for a jump process on a suitable discretiza- tion
of continuous conformation space. Here, we introduce statistical estimation
methods that allow a priori knowledge of equilibrium probabilities to be
incorporated into the estimation of dynamical observables. Both, maximum
likelihood methods and an improved Monte Carlo sampling method for reversible
transition ma- trices with fixed stationary distribution are given. The
sampling approach is applied to a toy example as well as to simulations of the
MR121-GSGS-W peptide, and is demonstrated to converge much more rapidly than a
previous approach in [F. Noe, J. Chem. Phys. 128, 244103 (2008)]Comment: 15 pages, 8 figure
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
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