Variational inference of fractional Brownian motion with linear computational complexity
This work provides an efficient method for researchers in fields like biophysics or finance needing to analyze stochastic processes from limited data, though it is incremental as it builds on existing likelihood-free and neural network techniques.
The authors tackled the problem of inferring parameters of fractional Brownian motion from single trajectories by developing a simulation-based, amortised Bayesian inference scheme with linear computational complexity, achieving precision scaling similarly to the Cramér-Rao bound across various trajectory lengths and robustness to positional noise.
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{é}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.