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

Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent H in (0,1) and a generalised diffusion coefficient D H > 0. We jointly estimate H and D H 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 D H 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 D H, the posterior mean has the lowest root mean squared error in both H and D H, except near H=0 and H=1 where MLE estimates H better. In vesicle trajectories from two cell types, H and D H 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.