quantum mechanics - Hubbard-Stratonovich transformation and mean-field approximation

For an interacting quantum system, Hubbard-Stratonovich transformation and mean-field field approximation are methods often used to decouple interaction terms in the Hamiltonian. In the first method, auxiliary fields are introduced via an integral identity, and then approximated by their saddle-point values. In the second method, operators are directly replaced by their mean values, e.g. $c_i^\dagger c_jc_k^\dagger c_l \rightarrow \langle c_i^\dagger c_j\rangle c_k^\dagger c_l + c_i^\dagger c_j \langle c_k^\dagger c_l\rangle$. In both methods, order parameters can then be solved self-consistently to yield the decoupled Hamiltonian.

Are these two methods equivalent? If not, how are they related?