Characteristic of Bennett’s acceptance ratio method

A powerful and well-established tool for free-energy estimation is Bennett’s acceptance ratio method. Central properties of this estimator, which employs samples of work values of a forward and its time-reversed process, are known: for given sets of measured work values, it results in the best estimate of the free-energy difference in the large sample limit. Here we state and prove a further characteristic of the acceptance ratio method: the convexity of its mean-square error. As a two-sided estimator, it depends on the ratio of the numbers of forward and reverse work values used. Convexity of its mean-square error immediately implies that there exists a unique optimal ratio for which the error becomes minimal. Further, it yields insight into the relation of the acceptance ratio method and estimators based on the Jarzynski equation. As an application, we study the performance of a dynamic strategy of sampling forward and reverse work values.

Figure 1

(Color online) The overlap density $p_{\alpha }\left(w\right)$ bridges the densities $p_0\left(w\right)$ and $p_1\left(w\right)$ of forward and reverse work values, respectively. $\alpha$ is the fraction $\frac{n_0}{n_0+n_1}$ of forward work values, here schematically shown for $\alpha =0.0001$, $\alpha =0.5$, and $\alpha =0.9999$. The accuracy of two-sided free-energy estimates depends on how good $p_{\alpha }\left(w\right)$ is sampled when drawing from $p_0\left(w\right)$ and $p_1\left(w\right)$.

Figure 2

The main figure displays the exponential work densities $p_0$ (thick line) and $p_1$ (thin line) for the choice of ${\mu }_0=10$ and, according to the fluctuation theorem, ${\mu }_1=10/11$. The inset displays the corresponding Boltzmann distributions ${\rho }_0\left(x,y\right)$ (thick) and ${\rho }_1\left(x,y\right)$ (thin) both for $y=0$. Here, ${\omega }_0$ is set equal to 1 arbitrarily, hence, ${\omega }_1^2=\left(1+{\mu }_0\right){\omega }_0^2=11$. The free-energy difference is $\Delta f=\text{ln}\left(1+{\mu }_0\right)=\text{ln}\left({\omega }_1^2/{\omega }_0^2\right)\approx 2.38$.

Figure 3

The overlap function $U_{\alpha }$ and the rescaled asymptotic mean-square error $M$ for ${\mu }_0=1000$. Note that $M\left(\alpha \right)$ diverges for $\alpha \to 0$.

Figure 4

The optimal fraction ${\alpha }_o=\frac{n_0}{N}$ of forward work values for the two-sided estimation in dependence of the average forward work ${\mu }_0$. For ${\mu }_0\le {\xi }_{+}\approx 3.56$, the one-sided forward estimator is optimal, i.e., ${\alpha }_o=1$.

Figure 5

(Color online) Example of a single run using the dynamic strategy: the optimal fraction ${\alpha }_o$ of forward measurements for the two-sided free-energy estimation is estimated at predetermined values of total sample sizes $N=n_0+n_1$ of forward and reverse work values. Subsequently, taking into account the current actual fraction $\alpha =\frac{n_0}{N}$, additional work values are drawn such that we come closer to the estimated ${\widehat{\alpha }}_o$.

Figure 6

(Color online) Displayed are estimated mean-square errors $\widehat{M}$ in dependence of $\alpha$ for different sample sizes. The global minimum of the estimated function $\widehat{M}$ determines the estimate of the optimal fraction ${\alpha }_o$ of forward work measurements.

Figure 7

(Color online) Comparison of a single run of free-energy estimation using the equal cost strategy versus a single run using the dynamic strategy. The error bars are the square roots of the estimated mean-square error $X$.

Figure 8

(Color online) Averaged estimates from $10000$ independent runs with dynamic strategy versus $10000$ runs with equal cost strategy in dependence of the total cost $c=n_0{c}_0+n_1{c}_1$ spend. The cost ratio is $c_1/c_0=0.01$, $c_0+c_1=2$, and ${\mu }_0=1000$. The error bars represent one standard deviation. Here, the initial value of $\alpha$ in the dynamic strategy is 0.5, while the equal cost strategy draws with ${\alpha }_{ec}\approx 0.01$. We note that ${\alpha }_o\approx 0.08$.

Figure 9

(Color online) Displayed are mean-square errors of free-energy estimates using the same data as in Fig. 8. In addition, the mean-square errors of estimates with constant $\alpha ={\alpha }_o$ are included, as well as the asymptotic behavior (51). The inset shows that the mean-square error of the dynamic strategy approaches the asymptotic optimum, whereas the equal cost strategy is suboptimal. Note that for small sample sizes, the asymptotic behavior does not represent the actual mean-square error.