Comments on Likelihood Fits with Variable Resolution

When several categories of events are present in the same data sample, an unbinned Maximum Likelihood fit is often used to determine the proportion and the properties of each class of events. This procedure makes use of “templates”, representing the probability distribution of the observables used in the fit for each class of events. In the simplest cases the templates are completely determined by the values assigned to the parameters of the fit, but frequently a more sophisticated approach is chosen where templates vary on an event by event basis, according to the resolution of the measurement for that particular event. These variations are due to the dependence of resolution on extra variables, that change on an event-by-event basis . This may happen, for instance, when events are recorded by a detector that has different resolutions in different regions within its acceptance. A common example of this kind of fit in HEP is given by lifetime and/or mass fits (see [1] for a sample list of recent experimental papers), where variations in resolution occur as a consequence of different configuration of each individual decay. The same kind of issue however is likely to arise in other situations. The purpose of this short paper is to point out some potential pitfalls in this kind of fitting procedure. I will illustrate the point with reference to a simple toy problem.