Overview of Non-Parametric Probability Density Estimation Methods

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

2. 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

3. 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

5. 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.

7. 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

8. Comparison of GEM and static-kernal PDE:

9. 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

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

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

12. PDE’s vs Neural Nets… • There are some problems that are particularly well suited to PDE’s:

13. PDE’s vs Neural Nets…

14. PDE’s vs Neural Nets…

15. PDE’s vs Neural Nets…

16. 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!