where S {\displaystyle S} is now the action of the bosonic fields, and M ( μ , σ ) {\displaystyle M(\mu ,\sigma )} is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable A [ σ ] {\displaystyle A[\sigma ]} is therefore an average over all configurations weighted by ρ [ σ ] {\displaystyle \rho [\sigma ]}