Phase transitions in the estimation of event rate: a path integral analysis

We try to capture the instantaneous rate of event occurrence as a function of time. A jagged or smooth rate function can be arbitrarily chosen, but it is nevertheless possible to select a plausible smoothness according to the principle of maximum likelihood. Upon optimization, the smoothness may diverge, indicating that the data are insufficient to uncover the time dependence of the underlying rate. By evaluating the likelihood function through a marginalization path integral, we found not only first-order but also second-order phase transitions leading to the divergence of the optimized smoothness. (fast track communication)