Working with a toy model whose partition function consists of a discrete summation, we introduce the
statistical field theory methodology by transforming a partition function via a formal Gaussian
integral relation (the Hubbard–Stratonovich transformation). We then consider Gaussian-type
approximations, wherein correlational contributions enter as harmonic fluctuations around the
saddle-point solution. This work focuses on how to arrive at a self-consistent, non-perturbative
approximation without recourse to a standard variational construction based on the
Gibbs–Bogolyubov–Feynman inequality that is inapplicable to a complex action. To address this
problem, we propose a construction based on selective satisfaction of a set of exact relations
generated by considering a dual representation of a partition function, in its original and
transformed form.