Title: Applying Neural Galerkin Normalizing Flows to Bayesian Inference in Diffusion Models with Inaccessible Boundaries

Abstract:

A major obstacle in performing Bayesian inference on diffusion model parameters using discrete data is the absence of a closed-form analytical solution for the transition density function between observation intervals. This function is essential for constructing the likelihood. Building on prior research that utilizes Normalizing Flows to address Fokker-Planck (FP) partial differential equations, this study introduces a novel Normalizing Flow architecture designed to approximate the transition density of diffusion processes across observation periods.

Our approach involves solving the corresponding FP equation within a Neural Galerkin framework, utilizing a Dirac mass as the initial condition. The training process is conducted over a defined distribution of initial states and diffusion coefficients. Particular attention is given to processes where the diffusion matrix becomes zero in specific inaccessible boundary zones, a characteristic observed in Stochastic Volatility models that adhere to the Feller condition.

By evaluating the learned transition densities along the observed trajectory, their product serves as an approximation of the likelihood function. This mechanism facilitates rapid posterior sampling through Markov Chain Monte Carlo (MCMC) methods. Once the offline training phase is complete, the inference process gains substantial efficiency. It eliminates the necessity of solving the FP equation in real-time for every parameter candidate suggested by the MCMC sampler, as well as the reliance on likelihood-free techniques that require repeated simulations of diffusion bridges.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC