Department of Chemical and Biomolecular Engineering, University of California-Berkeley, Berkeley, California 94720, USA.
J Chem Phys. 2013 Sep 28;139(12):121931. doi: 10.1063/1.4820491.
The evaluation of the Fisher information matrix for the probability density of trajectories generated by the over-damped Langevin dynamics at equilibrium is presented. The framework we developed is general and applicable to any arbitrary potential of mean force where the parameter set is now the full space dependent function. Leveraging an innovative Hermitian form of the corresponding Fokker-Planck equation allows for an eigenbasis decomposition of the time propagation probability density. This formulation motivates the use of the square root of the equilibrium probability density as the basis for evaluating the Fisher information of trajectories with the essential advantage that the Fisher information matrix in the specified parameter space is constant. This outcome greatly eases the calculation of information content in the parameter space via a line integral. In the continuum limit, a simple analytical form can be derived to explicitly reveal the physical origin of the information content in equilibrium trajectories. This methodology also allows deduction of least informative dynamics models from known or available observables that are either dynamical or static in nature. The minimum information optimization of dynamics is performed for a set of different constraints to illustrate the generality of the proposed methodology.
本文评估了在平衡态下由过阻尼朗之万动力学产生的轨迹概率密度的 Fisher 信息矩阵。我们所提出的方法具有通用性，适用于任何任意的平均力势，其中参数集现在是全空间相关函数。利用相应的福克-普朗克方程的创新厄米特形式，可以对时间传播概率密度进行本征基分解。这种表述促使我们使用平衡概率密度的平方根作为评估轨迹 Fisher 信息的基础，其主要优点是在指定参数空间中 Fisher 信息矩阵是常数。通过线积分，这一结果极大地简化了参数空间中信息量的计算。在连续极限下，可以推导出一个简单的解析形式，明确揭示平衡轨迹中信息量的物理起源。该方法还可以从已知或可用的、本质上是动态或静态的可观测量中推导出信息量最小的动力学模型。针对一组不同的约束条件，对动力学的最小信息优化进行了演示，以说明所提出方法的通用性。
