2 statistical analysis reweighting
to reweigh an observable we have to compute
∫dUe−S[U]O(U)r(U)∫dUe−S[U]r(U)=∫dUe−S[U]O(U)r(U)∫dUe−S[U]r(U)∫dUe−S[U]∫dUe−S[U]=⟨Or⟩⟨r⟩. r is computed stochastically as r(U)=⟨r(U)⟩ϕ=det(D(μf)/D(μi))=∫dϕew(U,ϕ) with w(U,ϕ)=ϕ†(1−D(μi)D−1(μf))ϕ and ϕ a gaussian noise P(ϕ)∝e−ϕ2.
Since det(D(μf)/D(μi)) may be complex we compute ∣det(D(μf)/D(μi))∣=√det(D−1(μi)D(μf)D(−μf)D−1(−μi))=(∫dϕeϕ†(1−D(−μi)D−1(−μf)D−1(μf)D(μi))ϕ)12 , thus the observable to compute is
⟨O⟨r⟩ϕ⟩U⟨⟨r⟩ϕ⟩U. 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=1Nϕ∑ϕr(Ui,ϕ)+O(1Nϕ) w(Ui)=log(1Nj∑jr(Ui,ϕij))=w(Ui,ϕi0)+log(1Nj∑jew(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(1Nj∑jr(Ui,ϕij))=12w(Ui,ϕi0)+12log(1Nj∑jew(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 ⟨O⟨r⟩ϕ⟩Ue−ˉw⟨⟨r⟩ϕ⟩Ue−ˉw. with ˉw=∑Uw(U). At this stage ⟨⟨r⟩ϕ⟩Ue−ˉw can be computed as ⟨⟨r⟩ϕ⟩Ue−ˉw=1NU∑Uew(U)−ˉw