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实现。在基于散射的分析中，研究人员常利用领域专业知识选择特定波矢范围进行分析，这种选择因具体案例而异。我们提出另一种范式：从数据处理之初就定义概率生成模型，并传播参数估计的不确定性，称之为从头不确定量化（AIUQ）。以差分动态显微镜（DDM）为例，该方法通过选定波矢范围内的傅里叶分析提取动力学信息。我们首先证明，DDM等价于在倒易空间中使用隐因子模型作为生成模型对时间变异函数进行拟合。随后推导出最大边际似然估计量，该估计量最优地加权所有波矢的信息，从而无需选择波矢范围。此外，我们利用托普利茨协方差的广义舒尔算法在无近似条件下大幅降低计算成本。模拟研究验证了AIUQ能显著提升估计精度，并通过自动化分析实现模型选择。三个不同实验组也展示了AIUQ的实用性：首先在各项同性牛顿流体中，相较于多粒子追踪，突破光学稠密系统的极限；其次在经历溶胶-凝胶转变的系统中，自动确定凝胶点和临界指数；最后在液晶中鉴别胶体各向异性扩散行为。这些成果共同凸显了AIUQ以高效自动化方式捕捉系统动力学的通用性。