Statistics for Particle Physics Lecture 2: Estimation

1
Statistics for Particle PhysicsLecture 2
Estimation

• Roger Barlow
• Manchester University
• TAE, Santander,10 July 2006

2
Estimation is part of Inferential statistics
Given these distribution parameters, what can we
say about the data?
Theory
Data
Given this data, what can we say about the
parameters of the distribution functions?
3
What is an estimator?
Often denoted by a caret (or hat)

• An estimator is a procedure giving a value for a
  parameter or property of the distribution as a
  function of the actual data values

V means variance. Its just ?2
4
What is a good estimator?
One often has to work with less-than-perfect
estimators

• A perfect estimator is
• Consistent
• Unbiassed
• Efficient
• minimum

5
Estimation Least Squares

• Measurements of y at various x with errors ? and
  prediction f(xa)

y
Minimise ?2
x
6
Least Squares The Really Nice Thing

• Should get ?2?1 per data point
• Minimise ?2 makes it smaller effect is 1 unit
  of ?2 for each variable adjusted. (Dimensionality
  of MultiD Gaussian decreased by 1.)
• Ndegrees Of FreedomNdata pts N parameters
• Provides Goodness of agreement figure which
  allows for credibility check

7
Chi Squared Results

• Large ?2 comes from
• Bad Measurements
• Bad Theory
• Underestimated errors
• Bad luck

• Small ?2 comes from
• Overestimated errors
• Good luck

8
The Straight Line

• You have points (xi, yi).
• xi known exactly. yi all have the same error ?
  (for now).
• Line ymxc
• Minimise ? (yi-mxi-c)2
• m xy-?x?y
• x2-?x2
• c?y-m?x

9
Errors on the straight line parameters

• m ?yi(xi- ?x )
• N (x2- ?x2)
• Error on m is given by combination of errors
  from errors on the yi
• (call them ?)
• ?m2? (xi- ?x )/N (x2- ?x2) 2 ? 2
• ? 2 .
• N (x2- ?x2)

10
Example fitting a number

• If you know signal and background distributions
  can fit for numbers
• Analyses often prefer to cut
• Why?

SIGNAL
11
How it works

• Count data events in signal region after cut ND
• Subtract (small?) number of background events.
  May be obtained is 2 ways
• Monte Carlo Simulation
• Only possibility before your see the data
• From the real data, outside the signal region.
• Often leave a gap, in case some signal leaks out
• Using a symmetric region makes linear effects
  cancel
• Number robust against Monte Carlo inadequacies
• Then say Ns ( ND NBack)/Effcy
• Where Effcy is the fraction of true data events
  that survive (have to get from MC)

12
Why cut rather than fit?

• Easy to implement cuts on several variables
• Reduces events to manageable numbers
• Gives an identifiable sample at the end
• Typical analysis will have many cuts. Some as
  early as the trigger. Filters, skims, user
  selections Final stage may involve fitting
  rather than cutting

13
How to choose your cuts (tuning)

• Signal size S after cuts,
• Background size B after cuts
• If you are looking for a new signal, optimise
  S/?B
• If you are measuring a (small) signal, optimise
  S/?(SB)
• (Requires prior assumptions about S)
• Tuning your cuts by looking at the data signal is
  BAD. It has led to fake discoveries in the past
  many experiments now have a policy of blind
  analysis.

14
The Michelangelo Technique
In 1501, Michelangelo returned to Florence, where
he was based until 1505. The major work of this
period is the colossal (4.34 m/143 ft) marble
David (1501-1504, Accademia, Florence). T he Old
Testament hero is depicted as a lithe, naked
youth, muscular and alert, looking into the
distance as if sizing up the enemy Goliath, whom
he has not yet encountered.  When sculpting this
statue Michelangelo is quoted to have expressed
that "When I saw this piece of marble, I knew
immediately that David was inside.  My job would
be to cut the excess marble away to reveal
him".  
15
Orthogonal Polynomials

• Fit a cubic Standard polynomial
• f(x)c0 c1x c2x2 c3x3
• Least Squares ?(yi-f(xi))2 gives

Invert and solve? Think first!
16
Define Orthogonal Polynomial

• P0(x)1
• P1(x)x a01P0(x)
• P2(x)x2 a12P1(x) a02P0(x)
• P3(x)x3 a23P2(x) a13P1(x) a03P0(x)
• Orthogonality ?rPi(xr) Pj(xr) 0 unless ij
• aij-(? ?r xrj Pi (xr))/ ?r Pi (xr)2

17
Use Orthogonal Polynomial

• f(x)c0P0(x) c1P1(x) c2P2(x) c3P3(x)
• Least Squares minimisation gives
• ci?yPi / ? Pi2
• Special Bonus These coefficients are
  UNCORRELATED
• Simple example
• Fit ymxc or
• ym(x -?x)c

18
Optimal Observables
g
f

• Function of the form
• P(x)f(x)a g(x)
• e.g. signalbackground, tau polarisation, extra
  couplings
• A measurement x contains info about a
• Depends on f(x)/g(x) ONLY.
• Work with O(x)f(x)/g(x)
• Write
• Use

x
O
19
Why this is magic
Its efficient. Saturates the MVB. As good as
ML x can be multidimensional. O is one
variable. In practice calibrate ?O and â using
Monte Carlo If a is multidimensional there is an
O for each ai If the form is quadratic then use
of the mean OO is not as good as ML. But close.
20
Example an Asymmetry

• (3/8)(1axx2) usually with x?Cos ?
• Can minimise
• (yi -(3/8)(1axixi2)) 2
• And obtain expressions for a and for ?a