Bayesian Field Theory of the Rate Estimation

We address the statistical inference of a time-dependent rate of events in the framework of Bayesian field theory. This maps the problem to a Langevin equation which, beyond the local linear regime taken as reference, involves nonlinearities and an explicit dependence on the local shape of the maximum likelihood curve. We study the corresponding impacts in a perturbative expansion, formulating a scaling hypothesis for the order of shape corrections. We find that the pure nonlinearities dominate the mean and skewness. Crucially, we uncover that the leading correction to the variance is driven by noise propagation from the signal's effective curvature. We test the derived expansion with numerical simulations and illustrate its applicability on real neural spike data.