The most common likelihood principle: the gaussian case G. Bonvicini, Wayne State University

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The most common likelihood principle the
gaussian caseG. Bonvicini, Wayne State University

• . With many thanks to Louis
• Talk based on paper to be published in NIM A
• For once, this is a talk where data
  configurations can be considered as smoothly
  distributed

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The principle

• P(aL)P(La)/G
• (L?f(L) a possibility), where
• L is the likelihood, L(ya)
• a are the fit parameters
• G is the generalized goodness of fit parameter,
  equals P(L)
• Defined in function space (typically not a
  Hilbert space)
• Clearly on a collision course with the maximum
  likelihood principle

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History and method

• Based on the likelihood theorem (Birnbaum 1963),
  and wishing to use all moments of the likelihood
  in the definition of G. Avoided the unbinned
  likelihood due to known problems there - only
  binned data.
• In practice one chooses a suitable set of
  moments or derivatives and develops the
  likelihood L ? (c2 -2lnLmax,m,s,) (more on it
  later). I choose derivatives and also develop
  f(L)lnL.

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Example all these four distributions fit to same
(c2,m,s) with Lu (J. Heinrich, PHYSTAT2003)
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Properties of the method

• The method works if strong correlation is
  observed at higher moments/derivatives
• The plots correlating two different moments tend
  to have a generally common shape

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Chi-square versus dm-a

• In the unbiased hypothesis, one finds that P(c2
  d0) peaks at c2Nbins, and
• Very low c2 values correspond to large d and/or
  large pulls

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sigma versus dm-a

• For poissonian and Gaussian statistics the
  distributions P(m,sa) do not merge smoothly into
  one another

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The method in the (m,s) plane

• A mapping takes place from the original
  likelihood (m,s) to the highest probability ones.
  The longest the distance traveled the worse the
  goodness of fit

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Likelihood with gaussian statistics (with
undergraduate Brandon Willard)

• -2lnL Si ((yi-fi(x,a))/si)2
• with substitutions
• ri (yi-fi(x,a))/si
• f(n)i f(n)i/si
• dm-a
• And
• P(a)gaussian with E(a)0 and D(a)Si a2i
• P(a,b)correlated gaussian with
  ?2(Saibi)/D(a)D(b)
• ?2(a,b)c2/D(a)D(b)

• -2lnL
• S ri2
• -2dfiri a
• d2 (fi2 -firi b)
• d3 (3fi fi -firi c)/3 .
• To second order P(La) P(aba)
• P(c2-2a2bab,a)

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A basic point about robustness correlation to
pull of the quantity you are trying to maximize
is paramount
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Conclusions

• A new method to fit data is being developed. It
  seeks to overcome the limitations of the
  likelihood to improve robustness. It provides a
  much stronger, generalized GOF parameter.
  Smaller corrections to peak locations and
  parameter uncertainties are also expected