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

$$\frac{\langle O \langle r\rangle_\phi \rangle_U}{\langle \langle r\rangle_\phi\rangle_{U}}\,.$$

1. Ignoring the error on $r$ we can define

   $$\bar r(U_i)=\frac{\sum_j r(U_i,\phi_{ij})}{\frac{1}{N_U}\sum_{i_1,j} r(U_{i_1},\phi_{ij})}$$ we can compute the sum in the numerator and in the denominator as in (Equation 2.1) and (Equation 2.2) then we can binning the data to take into account for autocorrelation $$\tilde O_b=\frac{1}{N_b}\sum_{i=bN_b}^{bN_b+N_b} O(U_i) \bar{r}(U_i)$$ and then the jackknifes as $$Jack(O)_j=\frac{1}{N-1}\sum_{a\neq j} \tilde O_a$$

2. to fully propagate the error instead we have to first to bin the data to take into account for autocorrelation $${ [Or]}_b=\frac{1}{N_b}\sum_{i=bN_b}^{bN_b+N_b} O(U_i) \frac{1}{N_j}\sum_jr(U_i,\phi_{ij})\\ [r]_b=\frac{1}{N_b}\sum_{i=bN_b}^{bN_b+N_b} \frac{1}{N_j}\sum_jr(U_i,\phi_{ij})$$ we can not compute $r(U_i,\phi_{ij})$ with double precision, but we can compute the exponent of it and the exponent of a sum $$w(U_i)=\log \left(\frac{1}{N_j}\sum_j r(U_i,\phi_{ij})\right)= w(U_i,\phi_{i0})+\log\left(\frac{1}{N_j}\sum_j e^{w(U_i,\phi_{ij})- w(U_i,\phi_{i0})}\right)\,.\\ \tag{2.1}$$ For the case of an OS reweighting we need to take into account the square root factor 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 compute the exponent of the binning. $P_b= \log([Or]_b)$ and $w_b=\log([r]_b)$ $$P_b=w(U_{bN_b}) +\log\left(\frac{1}{N_b}\sum_{i=bN_b}^{bN_b+N_b}O(U_i) e^{w(U_i)- w(U_{bN_b})}\right)\\ w_b=w(U_{bN_b}) +\log\left(\frac{1}{N_b}\sum_{i=bN_b}^{bN_b+N_b} e^{w(U_i)- w(U_{bN_b})}\right)$$ After we can compute the jackknifes of the ratio $$\frac{Jack(Or)_j}{Jack(r)_j}=\frac{\sum_{a\neq j} [O]_a }{\sum_{b\neq j} [r]_b}=\frac{\sum_{a\neq j} e^{P_a} }{\sum_{b\neq j} e^{w_{b}}}=\sum_{a\neq j}\frac{ 1 }{\sum_{b\neq j} e^{w_{b}-P_a}}$$

2.1 comparison

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