  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. We discuss how the error scales with the number
of windows. Then, we derive a central limit theorem for averages that are
obtained from umbrella sampling. The central limit theorem suggests an
estimator of the error contributions from individual windows, and we develop a
simple and computationally inexpensive procedure for implementing it. We
demonstrate this estimator for simulations of the alanine dipeptide and show
that it emphasizes low free energy pathways between stable states in comparison
to existing approaches for assessing error contributions. We discuss the
possibility of using the estimator and, more generally, the eigenvector method
for umbrella sampling to guide adaptation of the simulation parameters to
accelerate convergence.
