Bayesian field theory for the rate estimation

We consider the statistical inference of a time-dependent rate of events in the framework of Bayesian field theory. By mapping the problem to a stochastic partial differential equation, as it is standard approach in field theory, we are able to numerically sample the distribution around the maximum likelihood path. We then analytically consider the perturbative expansion around the local and linear solution, and at the leading-order correction to the variance we find novel terms which depend on the local shape of the maximum likelihood path. We show that this shape correction is statistically most important in the variance expansion than the nonlinearity corrections. We then demonstrate the general applicability of the simulation method by extending it to the case of uncertain event times and by estimating the mortality rate in Venice during the 1348 Black Death epidemics from indirect evidence.