Free energy evaluation from independent estimates and associated variances

As discussed in Sec. 6.3.2, during a SGE simulation, optimal weights are evaluated using Eq. 6.25, and only temporary values are obtained from Eq. 6.27. Therefore, for each optimal weight, the simulation produces a series of estimates, $\Delta f_1, \Delta f_2, \dots, \Delta f_P$. At a given time, the current value of $P$ depends, on average, on the time and on the update frequency of optimal weights. In this section, for convenience, the subscript in $\Delta f_i$ labels independent estimates. We also know that each $\Delta f_i$ value is affected by an uncertainty quantified by the associated variance $\delta^2(\Delta f_i)$ calculated via Eq. 6.26. We can then write $\hat{\Delta f}$, the optimal estimator of $P^{-1} \sum_{i=1}^P \Delta f_i$, by a weighted sum of the individual estimates[129]

$$\boxed{ \hat{\Delta f} = \frac{\sum_{i=1}^P [\delta^2(\Delta f_i)]^{-1} \Delta f_i} {\sum_{j=1}^P [\delta^2(\Delta f_j)]^{-1}}.}$$ (6.28)

Note that independent estimates with smaller variances get greater weight, and if the variances are equal then the estimator $\hat{\Delta f}$ is simply the mean value of the estimates. The uncertainty in the resulting estimate can be computed from the variances of the single estimates as

$$\delta^2(\hat{\Delta f}) = \left\{ \sum_{j=1}^P [\delta^2(\Delta f_j)]^{-1} \right\}^{-1}.$$ (6.29)

The ORAC program allows one to calculate $\hat{\Delta f}$ using either all available estimates or a fixed number of estimates, taken from the latest ones.