As multiplying α by a constant does not change the value of σ $^{2}$[ ¯ ζ ], we can use a constraint where 〈 α T ( x $^{′}$| x ) 〉$_{˜}$ π,π (and hence the denominator) is constant. The variance, Eq. 3, is then minimized using Lagrange multipliers, leading to the optimal α , Notably, α ∗ includes the sought quantity ζ . The estimator obtained by substituting this function into Eq. 2 can either be solved by self-consistent iteration, as originally described by Bennett [4], or by finding the zero of an implicit function obtained by rearrangement, which is similar to an expression described by Shirts et. al. [5]. The variance in ¯ ζ in Eq. 5 may be estimated by substituting the appropriate α into Eq. 3. By substituting α ∗ into Eq. 3, multiplying by a factor of unity, and separating the expression into two expectations, we obtain the following formula for the variance, which is analogous to an extant expression for the error in the Bennett’s method [5]. When samples are drawn from K replicas with integrands π$_{k}$ ( x ) and partition functions Z$_{k}$ for k = 1 , ..., K , the following generalized replica gas identity is applicable: Here, π$_{t}$ ( x ) and Z$_{t}$ are the targeted integrands and integrals, respectively. The sums run over all replicas which are not t . As in Eq. 2, the choice α$_{k}$ = 1 yields an identity originally described by Adib [3]. We now consider the optimization of α in the case where K > 2. The asymptotic variance 4 Z$_{t}$ = ∑ k = t 〈 α$_{k}$ π$_{t}$ ( x $^{′}$) 〉$_{π}$$_{k}$$_{,T}$$_{k}$ k = t 〈 α$_{k}$ T$_{k}$ ( x $^{′}$| x ) 〉$_{π}$$_{k}$$_{,π}$$_{t}$ . (8) σ $^{2}$[ ¯ ζ ] =   N$_{π}$ ∑ l =1 1 2 + 2 cosh ( ζ ln N$_{T}$ N$_{π}$ T ( x ′ $_{l}$| x$_{l}$ ) π ( x ′ $_{l}$) ) + N$_{T}$ ∑ m =1 1 2 + 2 cosh ( ζ ln N$_{T}$ N$_{π}$ T ( x ′ $_{m}$| x$_{m}$ ) π ( x ′ $_{m}$) )  $^{−}$$^{1}$ − 1 N$_{T}$ − 1 N$_{π}$ , (7) 1 + N$_{T}$ N$_{π}$ T ( x $^{′}$| x ) π ( x ′ ) e ζ 1 + N$_{T}$ N$_{π}$ T ( x ′ | x ) π ( x ′ $_{l}$) e $_{ζ}$, (6) N$_{π}$ ∑ l =1 1 1 + N$_{π}$ N$_{T}$ π ( x ′ $_{l}$) T ( x ′ $_{l}$| x$_{l}$ $_{)}$e − ¯ ζ − N$_{T}$ ∑ m =1 1 1 + N$_{T}$ N$_{π}$ T ( x ′ $_{m}$| x$_{m}$ ) π ( x ′ $_{m}$) e ¯ ζ = 0 , (5) α ∗ ∝ ( 1 N$_{T}$ π ( x $^{′}$) e − ζ + 1 N$_{π}$ T ( x $^{′}$| x ) ) − 1 . (4)