Hamiltonian Monte Carlo methods for spectroscopy data analysis

We introduce a Bayesian framework for confocal fluorescence spectroscopy that addresses key limitations of traditional fluorescence correlation spectroscopy. By jointly modeling molecular motion, optical effects, and photon detection, the method enables direct inference of molecular trajectories from photon count data. A superresolution parameter further refines trajectory estimation beyond the rate of data acquisition. To explore the resulting high-dimensional posterior distribution, we develop Hamiltonian Monte Carlo (HMC) algorithms adapted to the correlation structure of spectroscopy data. This structure gives rise to a stiff ordinary differential equation involving a two-scale discrete Laplacian. Analysis of its spectral properties yields a CFL-type stability condition for the commonly used Störmer–Verlet integrator, revealing instability at practical step sizes. We overcome this limitation with a semi-implicit (IMEX) scheme that separates stiff and non-stiff terms while exploiting the Laplacian's sparsity for efficiency. Numerical experiments demonstrate that IMEX maintains second-order accuracy and energy stability while permitting larger step sizes. This framework establishes a scalable foundation for future models of surface-constrained or multi-mode molecular motion.