Non-parametric Inference for Diffusion Processes: A Computational Approach via Bayesian Inversion for PDEs

TL;DR

This paper presents a theoretical and computational workflow for the non-parametric Bayesian inference of drift and diffusion functions of autonomous diffusion processes based on the partial differential equations arising from the infinitesimal generator of the underlying process.

Abstract

In this paper, we present a theoretical and computational workflow for the non-parametric Bayesian inference of drift and diffusion functions of autonomous diffusion processes. We base the inference on the partial differential equations arising from the infinitesimal generator of the underlying process. Following a problem formulation in the infinite-dimensional setting, we discuss optimization- and sampling-based solution methods. As preliminary results, we showcase the inference of a single-scale, as well as a multiscale process from trajectory data.

Figures (5)

• Figure 1: Data from Monte Carlo estimates of the first and second MFPT moments, together with the FEM solution for the true drift and diffusion functions. Colored intervals indicate pointwise $1.96$ standard deviations from the mean.
• Figure 2: Data from kernel density estimates of the pdf for $t=\{0.1, 0.5, 0.9\}$, together with the FEM solution for the true drift and diffusion functions. Colored intervals indicate pointwise $1.96$ standard deviations from the estimator.
• Figure 3: Top row: Posterior mean for the drift and log squared diffusion function, obtained through inference from MFPT moments data, and compared to the prior and exact solution. Colored intervals indicate $1.96$ standard deviations from the respective mean. Bottom row: Posterior mean predictive for the first two MFPT moments, compared to the prior mean predictive and the utilized data points.
• Figure 4: Posterior mean for the drift (left) and log squared diffusion (middle) function, obtained through inference from FP data, and compared to the prior and exact solution. Colored intervals indicate $1.96$ standard deviations from the respective mean. Posterior mean predictive at $t=0.9$ (right), compared to the prior mean predictive and the utilized data snapshot.
• Figure 5: Posterior mean for the drift (left) and log squared diffusion (middle) function, obtained through inference from multiscale FP data, and compared to the prior and exact solution for the coarse-grain model. Colored intervals indicate $1.96$ standard deviations from the respective mean. Posterior mean predictive at $t=0.9$ (right), compared to the prior mean predictive and the utilized data snapshot.