We present a scalable Bayesian framework for the analysis of confocal fluorescence spectroscopy data, addressing key limitations in traditional fluorescence correlation spectroscopy methods. Our framework captures molecular motion, microscope optics, and photon detection with high fidelity, enabling statistical inference of molecule trajectories from raw photon count data, introducing a superresolution parameter which further enhances trajectory estimation beyond the native time resolution of data acquisition. To handle the high dimensionality of the arising posterior distribution, we develop a family of Hamiltonian Monte Carlo (HMC) algorithms that leverages the unique characteristics inherent to spectroscopy data analysis. Here, due to the highly-coupled correlation structure of the target posterior distribution, HMC requires the numerical solution of a stiff ordinary differential equation containing a two-scale discrete Laplacian. By considering the spectral properties of this operator, we produce a CFL-type integrator stability condition for the standard St\"ormer-Verlet integrator used in HMC. To circumvent this instability we introduce a semi-implicit (IMEX) method which treats the stiff and non-stiff parts differently, while leveraging the sparse structure of the discrete Laplacian for computational efficiency. Detailed numerical experiments demonstrate that this method improves upon fully explicit approaches, allowing larger HMC step sizes and maintaining second-order accuracy in position and energy. Our framework provides a foundation for extensions to more complex models such as surface constrained molecular motion or motion with multiple diffusion modes.

翻译：我们提出了一种可扩展的贝叶斯框架，用于分析共聚焦荧光光谱数据，以解决传统荧光相关光谱方法的关键局限性。该框架高保真地捕捉分子运动、显微镜光学特性和光子检测过程，使得能够从原始光子计数数据中对分子轨迹进行统计推断，并引入了一个超分辨率参数，进一步提升了轨迹估计的精度，超越了数据采集的固有时间分辨率。为处理由此产生的高维后验分布，我们开发了一系列哈密顿蒙特卡洛算法，这些算法利用了光谱数据分析所固有的独特特性。在此，由于目标后验分布具有高度耦合的相关结构，HMC需要数值求解一个包含双尺度离散拉普拉斯算子的刚性常微分方程。通过分析该算子的谱特性，我们为HMC中使用的标准St\"ormer-Verlet积分器推导出了一个CFL类型的积分器稳定性条件。为规避此不稳定性，我们引入了一种半隐式方法，该方法对刚性和非刚性部分进行差异化处理，同时利用离散拉普拉斯算子的稀疏结构以提高计算效率。详细的数值实验表明，该方法优于完全显式方法，允许使用更大的HMC步长，并在位置和能量上保持二阶精度。我们的框架为扩展到更复杂的模型（如表面约束分子运动或多扩散模式运动）奠定了基础。