Web ensemble averages for retrieving relevant information from rejected Monte Carlo moves

We study the relevance of including information about rejected Monte-Carlo moves in path-sampling computations of free energies. For this purpose, we define webs as sets of paths linked by the path-sampling scheme and introduce an associated statistical ensemble. Within this web ensemble, we derive and test several statistical averages enabling to include information about configurational and path quantities belonging to the unselected trial moves. We numerically observe that retrieving this information does not always result in variance reduction, as theoretically predicted by Delmas and Jourdain. To explain the possible detrimental effect of information-retrieving from web sampling, an action for the webs is introduced. The behaviour of the statistical variance is observed to correlate to an overlapping area of a web action histogram. This area represents the probability that a generated web is such that the difference of its action between the targeted and reference ensembles is lower than the corresponding difference of free energy. Variance reductions are numerically observed for increased areas, as it is the case for the residence weight method proposed previously. More generally, web ensembles yield a rigorous framework for rationalizing existing methods and also for deriving potentially new methods enabling to retrieve relevant information from rejected trial moves.