Overview of Non-Parametric Probability Density Estimation Methods Sherry Towers State University of New York at Stony Brook

S.Towers All kernal PDF estimation methods (PDE’s) are developed from a simple idea… If a data point lies in a region where clustering of signal MC is tight, and bkgnd MC is loose, the point is likely to be signal

S.Towers  To estimate a PDF, PDE’s use the idea that any continuous function can be modelled by sum of some “kernal” function  Gaussian kernals are a good choice for particle physics  So, a PDF can be estimated by sum of multi-dimensional Gaussians centred about MC generated points

S.Towers  Best form of Gaussian kernal is a matter of debate:  Static-kernal PDE method uses a kernal with covariance matrix obtained from entire sample  The Gaussian Expansion Method (GEM), uses an adaptive kernal; the covariance matrix used for the Gaussian at each MC point comes from “local” covariance matrix.

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GEM vs Static-Kernal PDE  GEM gives unbiased estimate of PDF, but slower to use because local covariance must be calculated for each MC point  Static-kernal PDE methods have smaller variance, and are faster to use, but yield biased estimates of the PDF

S.Towers Comparison of GEM and static-kernal PDE:

S.Towers PDE vs Neural Networks  Both PDE’s and Neural Networks can take into account non-linear correlations in parameter space  Both methods are, in principle, equally powerful  For most part they perform similarly in an “average” analysis

S.Towers PDE vs Neural Networks  But, PDE’s have far fewer parameters, and algorithm is more intuitive in nature (easier to understand)

S.Towers Plus, PDE estimate of PDF can be visually examined:

S.Towers PDE’s vs Neural Nets…  There are some problems that are particularly well suited to PDE’s:

S.Towers PDE’s vs Neural Nets…

S.Towers PDE’s vs Neural Nets…

S.Towers PDE’s vs Neural Nets…

S.Towers Summary  PDE methods are as powerful as neural networks, and offer an interesting alternative  Very few parameters, easy to use, easy to understand, and yield unbinned estimate of PDF that user can examine in the multidimensional parameter space!