[论文解读] A theoretical interpretation of variance-based convergence criteria in perturbation-based theories

Xiao-Hui Wang, Zhaoxi Sun|arXiv (Cornell University)|Feb 22, 2018

Advanced Thermodynamics and Statistical Mechanics参考文献 68被引用 27

一句话总结

本文為基於方差的收斂標準在擾動法自由能計算中的理論基礎，顯示FEP與BAR等估計器對自由能估計的方差具有非線性依賴性。研究證明，由於樣本數限制與分佈假設，基於方差的收斂指標本質上存在偏倚，導致不確定性被系統性低估——這使得實際應用中標準誤差估計毫無意義。

ABSTRACT

In QM/MM indirect free energy simulation, QM/MM corrections can be obtained from integration of partial derivatives of alchemical Hamiltonians or from perturbation-based estimators including free energy perturbation (FEP) and acceptance ratio methods. With FEP or exponential averaging, researchers tend to only sample MM states and calculate single point energy to get the free energy estimates. In this case the sample size hysteresis arises and the convergence is determined by bias elimination rather than variance minimization. Various criteria are proposed to evaluate the convergence issue and numerical studies are reported. It has been found that criteria including variance of distribution, effective sample size, information entropies and so on can be used and they are variance-of-distribution-dependent. However, no theoretical interpretation is presented. In this paper we present theoretical interpretations to dig the underlying statistical nature behind the problem. The convergence criteria are proven to be related with variance of distribution in Gaussian approximated Exponential averaging. Further, we prove that these estimators are nonlinearly dependent on the variance of the free energy estimate. As these estimators are often orders of magnitude overestimated, the variance of the FEP estimate is orders of magnitude underestimated. Hence, computing this statistical uncertainty is meaningless. In numerical calculation from timeseries data the effective sample size is bounded by 1 and N and thus the variance of the free energy estimate is proven to be bounded by 0 and 1 (kBT)2 for EXP and 0 and 2 (kBT)2 for BAR, which indicates an inevitable underestimation. Specifically, the upper bounds for these estimators are sample-size dependent. The effective sample size is proven to be a function of the overlap scalar, from which the range of the overlap scalar can also be derived.

研究动机与目标

• 理論闡釋擾動法自由能方法中基於方差的收斂標準的統計性質。
• 識別FEP與BAR計算中基於方差的估計器系統性偏倚的根本原因。
• 證明自由能估計中的標準誤差因有效樣本數受限與分佈假設而本質上不可靠。
• 澄清為何常見的收斂指標（如方差、有效樣本數與熵）在實務中具有欺騙性。
• 推導在指數平均與接受率平均下，自由能估計方差的理論邊界。

提出的方法

• 在高斯近似下，推導自由能估計方差與指數平均中收斂標準之間的理論關係。
• 證明FEP與BAR估計器對自由能估計方差具有非線性依賴性，導致對精確度的高估。
• 分析時間序列資料，顯示有效樣本數介於1與N之間，從而將自由能估計的方差約束在EXP的0至(kBT)²之間，以及BAR的0至2(kBT)²之間。
• 引入重疊標量作為決定有效樣本數的關鍵參數，並推導其理論範圍。
• 運用數學推導與漸近分析，建立自由能估計器統計不確定性的上界。
• 透過數值示範驗證理論預測，包括在最終版本中新增的圖7（Figure 7）。

实验结果

研究问题

• RQ1 為何FEP與BAR中基於方差的收斂標準經常無法反映真實的統計不確定性？
• RQ2 自由能估計方差與實務中使用的收斂指標之間的理論關係為何？
• RQ3 有效樣本數如何約束時間序列資料中自由能估計的方差？
• RQ4 為何擾動法中自由能估計的標準誤差在系統上被低估？
• RQ5 重疊標量在決定收斂標準可靠性方面扮演何種角色？

主要发现

• 指數平均中自由能估計的方差介於0與(kBT)²之間，表示真實不確定性不可能超過此限界。
• 對於BAR方法，自由能估計的方差介於0與2(kBT)²之間，顯示統計不確定性的根本上限。
• 時間序列中有效樣本數介於1與N之間，此一性質本質上限制了基於方差的收斂標準的精確度。
• 重疊標量決定了有效樣本數的範圍，其理論邊界已由分析推導得出。
• 基於方差的收斂標準對真實方差具有非線性依賴性，導致對精確度的高估達數個數量級，同時系統性低估實際不確定性。
• 因此，由於系統性偏倚與方差受限，從這些估計器計算標準誤差在統計上毫無意義。