Loading [MathJax]/jax/output/SVG/jax.js

2  statistical analysis reweighting

to reweigh an observable we have to compute

dUeS[U]O(U)r(U)dUeS[U]r(U)=dUeS[U]O(U)r(U)dUeS[U]r(U)dUeS[U]dUeS[U]=Orr. r is computed stochastically as r(U)=r(U)ϕ=det(D(μf)/D(μi))=dϕew(U,ϕ) with w(U,ϕ)=ϕ(1D(μi)D1(μf))ϕ and ϕ a gaussian noise P(ϕ)eϕ2.

Since det(D(μf)/D(μi)) may be complex we compute det(D(μf)/D(μi))=det(D1(μi)D(μf)D(μf)D1(μi))=(dϕeϕ(1D(μi)D1(μf)D1(μf)D(μi))ϕ)12 , thus the observable to compute is

OrϕUrϕU. we can not compute r(Ui,ϕij) with double precision, but we can compute the exponent of its sum over ϕ as ew(Ui)=r(Ui,ϕ)ϕ=1Nϕϕr(Ui,ϕ)+O(1Nϕ) w(Ui)=log(1Njjr(Ui,ϕij))=w(Ui,ϕi0)+log(1Njjew(Ui,ϕij)w(Ui,ϕi0)).(2.1) For the case of an OS reweighting we need to take into account the square root factor ew(Ui)=(r(Ui,ϕ)ϕ)1/2 as w(Ui)=12log(1Njjr(Ui,ϕij))=12w(Ui,ϕi0)+12log(1Njjew(Ui,ϕij)w(Ui,ϕi0)).(2.2)

then we can multiply numerator and denominator by a factor to make the computation doable in double precision OrϕUeˉwrϕUeˉw. with ˉw=Uw(U). At this stage rϕUeˉw can be computed as rϕUeˉw=1NUUew(U)ˉw