One of the classic data mining tasks is to discover bursts, time intervals, where events occur at abnormally high rate. In this paper we revisit Kleinberg's seminal work, where bursts are discovered by using exponential distribution with a varying rate parameter: the regions where it is more advantageous to set the rate higher are deemed bursty. The model depends on two parameters, the initial rate and the change rate. The initial rate, that is, the rate that is used when there are no burstiness was set to the average rate over the whole sequence. The change rate is provided by the user. We argue that these choices are suboptimal: it leads to worse likelihood, and may lead to missing some existing bursts. We propose an alternative problem setting, where the model parameters are selected by optimizing the likelihood of the model. While this tweak is trivial from the problem definition point of view, this changes the optimization problem greatly. To solve the problem in practice, we propose efficient ($1 + ε$) approximation schemes. Finally, we demonstrate empirically that with this setting we are able to discover bursts that would have otherwise be undetected.