# Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters

arXiv: https://arxiv.org/abs/2610.19888 · HTML: https://mlsanity.com/paper/2610.19888/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 84 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 6 of 6)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
- [Neural Networks for Parameter Estimation in Intractable Models](https://mlsanity.com/paper/2107.14346/) (2021)
- [Maximum Likelihood Estimation for Single Particle, Passive Microrheology Data with Drift](https://mlsanity.com/paper/1509.03261/) (2015)

## Abstract (arXiv)

Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent H ∈ (0,1) and a generalised diffusion coefficient DH > 0. We jointly estimate H and DH using maximum likelihood (MLE) from density measurements with Poisson counting noise and single-particle trajectory with Gaussian localisation error. We show that one density measurement in time cannot infer both H and DH and that several time points are needed, which matters most for inference precision. For trajectories, we show that a neural network trained with a squared error loss approximates the posterior mean of H, which can be computed exactly at negligible cost. Paired with the profile-likelihood estimate of DH, the posterior mean has the lowest root mean squared error in both H and DH, except near H=0 and H=1 where MLE estimates H better. In vesicle trajectories from two cell types, H and DH form a continuum rather than distinct modes of motion. The posterior probability that each window within a track is subdiffusive or superdiffusive shows that 26% of MRC5 fibroblast and 21% of HeLa vesicle trajectories contain both subdiffusive and superdiffusive windows, against 2% and 7% expected from estimation uncertainty alone. Intracellular cargoes therefore switch between subdiffusion and superdiffusion over time rather than forming populations with fixed H.