1. In regions where dE/dx gives usable PID information, this information is desired in the form of probabilities for each of the possible identities associated with a track.
2. ID probabilities obviously can depend on the relative population of adjacent dE/dx bands, and relative populations depend on many possible cuts and parameters that necessarily vary from one analysis to another. Therefore, no pre-calibrated parameters are used, and the algorithm requires an initial calibration pass on an adequately large set of tracks. Of course, parameters generated in a calibration pass can be saved if desired, and reloaded later.
3. There needs to be provision for the variation of dE/dx band populations with p/Z, with  p_T /Z, and with N_hits, and for the variation of the widths of dE/dx bands with N_hits.

This code extracts ID probabilities beginning from the standard approach of fitting Gaussians to lines of pixels in the plane of dE/dx vs. p/Z. In order to minimize the number of free parameters and to make the fitting as stable and robust as possible, we use known information about the dE/dx bands to the maximum extent possible. In particular, we know that the mean of a band follows the Bethe-Bloch curve (albeit modified for detector effects); we also know the behavior of its rms width, when properly corrected for binning effects. Furthermore, the population of a band (i.e., the normalization of the Gaussian) varies smoothly with p/Z.

The chosen approach for handling  p_T  dependence and N_hits dependence relies on user-defined bins. The chosen  p_T  bins do not need to have a single fixed width. If the user does not specify binning for one or both of these variables, then we average over that variable when generating ID parameters. If binning in one or both is specified, the code finds the bin with the largest population of tracks and determines a set of parameters in the plane of dE/dx vs.  p/Z, as described below. For the present, it is assumed that the Bethe-Bloch curves do not vary with  p_T  or N_hits. It is also assumed that the widths of dE/dx bands scale like 1/N_hits^0.46, and do not vary with  p_T . Therefore, only the normalizations of the Gaussians are refit when stepping over the bins of  p_T  and N_hits.

In the first plane of dE/dx vs.  p/Z, the mean, the width and the population of dE/dx bands are fitted in relatively "clean" regions of the dE/dx vs.  p/Z plane, then extrapolated or interpolated into the more confused regions. The philosophy is to use these extrapolations/interpolations as initial values for a new fit wherever possible. This approach is robust enough to give reliable ID probabilities throughout the region, for example, where the electron band crosses the low momentum part of the pion band. In this instance, parameters in the overlap region are quite reliable because they are interpolations rather than extrapolations. The other important overlap regions are where the kaon band merges into the pion band just above 0.6 GeV/c, and where the proton band merges with the pions and kaons above 1 GeV/c. In these two instances, there is a narrow "semi-overlap" region where the two populations can be independently refit. Beyond that, there is a "complete overlap" region where the only parameter that can be refit represents the sum of two Gaussians with almost the same mean and width. In this "complete overlap" region, the code can provide relative probabilities, but these are parametrization-dependent extrapolations and are obviously not as reliable as the normal ID probabilities.