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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 , 

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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, 

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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, 

<img src='content_image/352104.jpg'>

and separating the expression into two expectations, we obtain the following formula for the variance, 

<img src='content_image/352103.jpg'>

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: 

<img src='content_image/352102.jpg'>

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 

