Overview of Non-Parametric

Probability Density Estimation Methods

Sherry TowersState 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.

S.Towers

S.Towers

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!