Choose a trial wave function \( \psi_T(\boldsymbol{R}) \). $$ P(\boldsymbol{R},\boldsymbol{\alpha})= \frac{\left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2}{\int \left|\psi_T(\boldsymbol{R},\boldsymbol{\alpha})\right|^2d\boldsymbol{R}}. $$ This is our model, or likelihood/probability distribution function (PDF). It depends on some variational parameters \( \boldsymbol{\alpha} \). The approximation to the expectation value of the Hamiltonian is now $$ \langle E[\boldsymbol{\alpha}] \rangle = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R},\boldsymbol{\alpha})\Psi_T(\boldsymbol{R},\boldsymbol{\alpha})}. $$