1,758 research outputs found
Control Variates for Reversible MCMC Samplers
A general methodology is introduced for the construction and effective
application of control variates to estimation problems involving data from
reversible MCMC samplers. We propose the use of a specific class of functions
as control variates, and we introduce a new, consistent estimator for the
values of the coefficients of the optimal linear combination of these
functions. The form and proposed construction of the control variates is
derived from our solution of the Poisson equation associated with a specific
MCMC scenario. The new estimator, which can be applied to the same MCMC sample,
is derived from a novel, finite-dimensional, explicit representation for the
optimal coefficients. The resulting variance-reduction methodology is primarily
applicable when the simulated data are generated by a conjugate random-scan
Gibbs sampler. MCMC examples of Bayesian inference problems demonstrate that
the corresponding reduction in the estimation variance is significant, and that
in some cases it can be quite dramatic. Extensions of this methodology in
several directions are given, including certain families of Metropolis-Hastings
samplers and hybrid Metropolis-within-Gibbs algorithms. Corresponding
simulation examples are presented illustrating the utility of the proposed
methods. All methodological and asymptotic arguments are rigorously justified
under easily verifiable and essentially minimal conditions.Comment: 44 pages; 6 figures; 5 table
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
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
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