Understanding the sources of error in MBAR through asymptotic analysis.

Many sampling strategies commonly used in molecular dynamics, such as umbrella sampling and alchemical free energy methods, involve sampling from multiple states. The Multistate Bennett Acceptance Ratio (MBAR) formalism is a widely used way of recombining the resulting data. However, the error of the MBAR estimator is not well-understood: previous error analyses of MBAR assumed independent samples.

Stratification as a general variance reduction method for Markov chain Monte Carlo.

The Eigenvector Method for Umbrella Sampling (EMUS) [46] belongs to a popular class of methods in statistical mechanics which adapt the principle of stratified survey sampling to the computation of free energies. We develop a detailed theoretical analysis of EMUS. Based on this analysis, we show that EMUS is an efficient method for computing averages over arbitrary target distributions.

Trajectory stratification of stochastic dynamics.

We present a general mathematical framework for trajectory stratification for simulating rare events. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of state space (strata), computing averages over the distributions of the trajectory fragments within the strata with minimal communication between them, and combining those averages with appropriate weights to yield averages with respect to the original underlying process. Our framework reveals the full generality and flexibility of trajectory stratification, and it illuminates a common mathematical structure shared by existing algorithms for sampling rare events.

Eigenvector method for umbrella sampling enables error analysis.

Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we introduce a mathematical framework that casts the step of combining the data as an eigenproblem. The advantage to this approach is that it facilitates error analysis.

SHARP ENTRYWISE PERTURBATION BOUNDS FOR MARKOV CHAINS.

For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem in computational statistical physics. We have derived perturbation bounds on the relative error of the invariant distribution that reveal these variations in sensitivity. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature.

A robust measure of correlation between two genes on a microarray.

Background: The underlying goal of microarray experiments is to identify gene expression patterns across different experimental conditions. Genes that are contained in a particular pathway or that respond similarly to experimental conditions could be co-expressed and show similar patterns of expression on a microarray. Using any of a variety of clustering methods or gene network analyses we can partition genes of interest into groups, clusters, or modules based on measures of similarity.