Empirical Bayes interpretations of random point events

Given a sequence of apparently random point events, such as neuronal spikes, one may interpret them as being derived either irregularly in time from a constant rate or regularly from a fluctuating rate. To determine which interpretation is more plausible in any given case, we employ the empirical Bayes method. For a sequence of point events derived from a rate-fluctuating gamma process, the marginal likelihood function can possess two local maxima and the system exhibits a first-order phase transition representing the switch of the most plausible interpretation from one to the other.