In order to bring in the Monte Carlo machinery, we define first a likelihood distribution, or probability density distribution (PDF). Using our ansatz for the trial wave function \( \psi_T(\boldsymbol{R};\boldsymbol{\alpha}) \) we define a PDF
$$ P(\boldsymbol{R})= \frac{\left|\psi_T(\boldsymbol{R};\boldsymbol{\alpha})\right|^2}{\int \left|\psi_T(\boldsymbol{R};\boldsymbol{\alpha})\right|^2d\boldsymbol{R}}. $$This is our model for probability distribution function. The approximation to the expectation value of the Hamiltonian is now
$$ \overline{E}[\boldsymbol{\alpha}] = \frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})H(\boldsymbol{R})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})} {\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R};\boldsymbol{\alpha})\Psi_T(\boldsymbol{R};\boldsymbol{\alpha})}. $$