Many applications in Lattice field theory require to determine the Taylor
series of observables with respect to action parameters. A primary example is
the determination of electromagnetic corrections to hadronic processes. We show
two possible solutions to this general problem, one based on reweigting, that
can be considered a generalization of the RM123 method. The other based on the
ideas of Numerical Stochastic Perturbation Theory (NSPT) in the Hamiltonian
formulation. We show that 1) the NSPT-based approach shows a much reduced
variance in the determination of the Taylor coefficients, and 2) That both
approaches are related by a change of variables. Numerical results are shown for
the case of $\lambda-\phi^4$ in 4 dimensions, but we expect these observations to be
general. We conclude by commenting on the possible use of Machine Learning
techniques to find similar change of variables that can potentially reduce the
variance in Taylor coefficients.