A Bayesian analysis of Green’s function Monte Carlo correlation functions

Imaginary-time correlation functions calculated by quantum Monte Carlo (QMC) are analyzed using the maximum entropy method (MaxEnt) to determine the ground-state energy and spectral overlap function. In contrast to earlier applications of MaxEnt, the data is obtained from importanced-sampled zero-temperature quantum Monte Carlo simulations. The analysis includes two steps. First, that spectral overlap function and ground state energy which maximizes the entropy and agrees with the QMC correlation functions is obtained. Then the errors in the energy are evaluated by averaging over all the possible images (average MaxEnt method), the multidimensional integrals being computed using the Metropolis algorithm. The central feature of this approach is that all the information present in the correlation functions is used in the only way consistent with fundamental probabilistic hypotheses. This allows us to fully exploit the information contained in the correlation functions at small imaginary times, thus avoiding large statistical fluctuations associated with large imaginary times. In addition, the computed errors include both the statistical errors and systematic extrapolation errors. The method is illustrated with a harmonic oscillator and the four-electron LIH molecule.