Variational inference of fractional Brownian motion with linear computational complexity

TL;DR

This paper presents a novel, likelihood-free Bayesian inference method using neural networks to efficiently estimate parameters of fractional Brownian motion from single trajectories, with linear computational complexity and high accuracy.

Contribution

It introduces a simulation-based amortized inference scheme combining graph neural networks and invertible neural networks for parameter estimation of random walks.

Findings

Linear scaling of computational complexity with trajectory length

High precision close to Cramér-Rao bound across various lengths

Robust inference in noisy environments and for finite decorrelation times

Abstract

We introduce a simulation-based, amortised Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learnt summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cram{\'e}r-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes well to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.