2  statistical analysis reweighting

to reweigh an observable we have to compute

\frac{\int dU e^{-S[U]} O(U) r(U)}{\int dU e^{-S[U]}r(U)}= \frac{\int dU e^{-S[U]} O(U) r(U)}{\int dU e^{-S[U]}r(U)} \frac{\int dU e^{-S[U]}}{\int dU e^{-S[U]}} = \frac{\langle Or \rangle}{\langle r\rangle}\,. r is computed stochastically as r(U)= \langle r(U) \rangle_\phi= \det\left(D(\mu_f)/D(\mu_i)\right) = \int d\phi e^{w(U,\phi)} with w(U,\phi)=\phi^\dagger(1-D(\mu_i) D^{-1}(\mu_f))\phi and \phi a gaussian noise P(\phi)\propto e^{-\phi^2}.

Since \det\left(D(\mu_f)/D(\mu_i)\right) may be complex we compute \left| \det\left(D(\mu_f)/D(\mu_i)\right)\right|=\sqrt{\det\left(D^{-1}(\mu_i)D(\mu_f)D(-\mu_f)D^{-1}(-\mu_i)\right)}\\ =\left(\int d\phi e^{\phi^\dagger(1-D(-\mu_i)D^{-1}(-\mu_f)D^{-1}(\mu_f)D(\mu_i))\phi}\right)^{\frac{1}{2}} , thus the observable to compute is

\frac{\langle O \langle r\rangle_\phi \rangle_U}{\langle \langle r\rangle_\phi\rangle_{U}}\,. we can not compute r(U_i,\phi_{ij}) with double precision, but we can compute the exponent of its sum over \phi as e^{w(U_i)}=\langle r(U_i,\phi)\rangle_\phi=\frac{1}{N_\phi}\sum_\phi r(U_i,\phi)+O\left(\frac{1}{N_\phi}\right) w(U_i)=\log \left(\frac{1}{N_j}\sum_j r(U_i,\phi_{ij})\right)= w(U_i,\phi_{i0})+\log\left(\frac{1}{N_j}\sum_j e^{w(U_i,\phi_{ij})- w(U_i,\phi_{i0})}\right)\,.\\ \tag{2.1} For the case of an OS reweighting we need to take into account the square root factor e^{w(U_i)}=\left(\langle r(U_i,\phi)\rangle_\phi\right)^{1/2} as w(U_i)=\frac{1}{2}\log \left(\frac{1}{N_j}\sum_j r(U_i,\phi_{ij})\right)= \frac{1}{2}w(U_i,\phi_{i0})+\frac{1}{2}\log\left(\frac{1}{N_j}\sum_j e^{w(U_i,\phi_{ij})- w(U_i,\phi_{i0})}\right)\,.\\ \tag{2.2}

then we can multiply numerator and denominator by a factor to make the computation doable in double precision \frac{\langle O \langle r\rangle_\phi \rangle_U e^{-\bar w}}{\langle \langle r\rangle_\phi\rangle_{U}e^{-\bar w}}\,. with \bar w = \sum_U w(U). At this stage \langle \langle r\rangle_\phi\rangle_{U}e^{-\bar w} can be computed as \langle \langle r\rangle_\phi\rangle_{U}e^{-\bar w} =\frac{1}{N_U} \sum_U e^{w(U)-\bar w}