Estimating parameters from data is a fundamental problem in physics, customarily done by minimizing a loss function between a model and observed statistics. In scattering-based analysis, researchers often employ their domain expertise to select a specific range of wavevectors for analysis, a choice that can vary depending on the specific case. We introduce another paradigm that defines a probabilistic generative model from the beginning of data processing and propagates the uncertainty for parameter estimation, termed ab initio uncertainty quantification (AIUQ). As an illustrative example, we demonstrate this approach with differential dynamic microscopy (DDM) that extracts dynamical information through Fourier analysis at a selected range of wavevectors. We first show that DDM is equivalent to fitting a temporal variogram in the reciprocal space using a latent factor model as the generative model. Then we derive the maximum marginal likelihood estimator, which optimally weighs information at all wavevectors, therefore eliminating the need to select the range of wavevectors. Furthermore, we substantially reduce the computational cost by utilizing the generalized Schur algorithm for Toeplitz covariances without approximation. Simulated studies validate that AIUQ significantly improves estimation accuracy and enables model selection with automated analysis. The utility of AIUQ is also demonstrated by three distinct sets of experiments: first in an isotropic Newtonian fluid, pushing limits of optically dense systems compared to multiple particle tracking; next in a system undergoing a sol-gel transition, automating the determination of gelling points and critical exponent; and lastly, in discerning anisotropic diffusive behavior of colloids in a liquid crystal. These outcomes collectively underscore AIUQ's versatility to capture system dynamics in an efficient and automated manner.

翻译：从数据中估计参数是物理学中的一个基本问题，通常通过最小化模型与观测统计量之间的损失函数来完成。在基于散射的分析中，研究者常利用其领域专业知识选择特定的波矢范围进行分析，这一选择可能因具体情况而异。我们引入另一种范式，该范式从数据处理之初就定义一个概率生成模型，并为参数估计传播不确定性，称为从头算不确定性量化（AIUQ）。作为一个示例，我们通过微分动态显微术（DDM）来演示这种方法，DDM通过在选定的波矢范围内进行傅里叶分析来提取动力学信息。我们首先证明，DDM等价于在倒易空间中使用潜因子模型作为生成模型来拟合时间变异函数。然后，我们推导出最大边际似然估计量，该估计量能最优地权衡所有波矢处的信息，从而无需选择波矢范围。此外，我们通过利用广义Schur算法处理Toeplitz协方差矩阵而无需求近似，大幅降低了计算成本。模拟研究验证了AIUQ能显著提高估计精度，并实现自动化分析下的模型选择。AIUQ的实用性也通过三组不同的实验得到证明：首先是在各向同性牛顿流体中，与多粒子追踪相比，突破了光学稠密系统的极限；其次是在经历溶胶-凝胶转变的系统中，自动确定了凝胶点和临界指数；最后，在辨别液晶中胶体各向异性扩散行为方面。这些结果共同凸显了AIUQ以高效、自动化方式捕捉系统动力学的多功能性。