Overview of Non-Parametric Probability Density Estimation Methods

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Overview of Non-Parametric Probability Density
Estimation Methods

• Sherry Towers
• State University of New York
• at Stony Brook

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• All kernal PDF estimation methods (PDEs) 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

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• To estimate a PDF, PDEs 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

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• 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

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Comparison of GEM and static-kernal PDE

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PDE vs Neural Networks

• Both PDEs 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

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PDE vs Neural Networks

• But, PDEs have far fewer parameters, and
  algorithm is more intuitive in nature (easier to
  understand)

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Plus, PDE estimate of PDF can be visually
examined

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PDEs vs Neural Nets

• There are some problems that are particularly
  well suited to PDEs

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PDEs vs Neural Nets

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PDEs vs Neural Nets

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PDEs vs Neural Nets

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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!