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```LHC-HCG meeting
Matters of convention
for “expected” results
1. Mean vs. median vs. Asimov
2. How do we define 1s and 2s bands ?
Mingshui Chen
Jan 27, 2011 Mingshui Chen ( University of Florida )
1
A few examples
Differential distributions of CL.95% upper limits (Bayesian
with flat prior) for 1000 sets of outcomes
1. Large statistics
nbkg=100, nsig=50, no systematics
Discreteness of lines hardly
seen
2. Single channel with
small statistics
3. Multiple channels
with small statistics
nbkg=1, nsig=1, no systematics
The discrete probabilities will be
changing smoothly as one moves
along r from small to large r
two channel, no systematics
nsig1=0.3 nbkg1=3 nsig2=0.5 nbkg2=1
The discrete probabilities will
NOT be changing smoothly as
one moves along r from small
to large r, due to the dips
Jan 27, 2011 Mingshui Chen ( University of Florida )
2
Mean or Median or “Asimov”
Do we show "mean"
well defined, requires almost no conventions
but statisticians do not like "mean" as it is not “preserved”
under transformations of variables; but do we really care?
or "median"
requires a convention for highly discrete distributions
or “Asimov data" ?
“imaginary” (in general, non-integer) most probable
experimental outcome
requires a whole paragraph of explanations
Jan 27, 2011 Mingshui Chen ( University of Florida )
3
Conventions of bands
A simple naïve convention
±1 and ±2 standard deviation (“two sided”) spread of the expectation obtained from a
large number of toy experiments
It becomes rather problematic for small statistics cases where bands must be
asymmetric and even one-sided…
Build up CL intervals by adding up probability-ranked possible
experimental outcomes
If p(r) changes smoothly, say first rising up and then falling down, then one would get a
continuous interval.
If probabilities are jumping up and down, such procedure would give disjoint sections as
in the case of multiple channels with small statistics
Build up CL intervals following “quantiles” using cumulative distributions
see next slides
Build up CL intervals following minimum-range principle
see next slides
We need a well-defined convention
Jan 27, 2011 Mingshui Chen ( University of Florida )
4
Convention of bands in LandS
Throw 1000 toy experiments according to the background-only model
(use bkgd systematic errors when non-zero)
Evaluate r95% for each of the 1000 toy experiments
Make differential and cumulative distributions of the obtained r95% values
differential distribution of r95%
cumulative distribution of r95%
0.977
0.841
0.5 (median)
0.159
0.023
average expected <r>
Jan 27, 2011 Mingshui Chen ( University of Florida )
To define median/bands, use crossings of the percentile lines
and the interpolation line between nearby pairs of physically
possible values of r. The current interpolation choice is the
Fermi function
5
Bands with different interpolations
nbkg=1, nsig=1, no systematics, Bayesian with flat prior
Fermi function interpolation
- 2s
= 3.00
- 1s
= 3.00
median = 3.45
+1s
= 4.73
+2s
= 6.67
Jan 27, 2011 Mingshui Chen ( University of Florida )
Linear interpolation
- 2s
= 3.00
- 1s
= 3.00
median = 3.46
+1s
= 4.88
+2s
= 6.72
Step function
- 2s
= 3.00
- 1s
= 3.00
median = 4.11
+1s
= 5.41
+2s
= 6.78
6
Bands with minimum-range principle
E.L. Crow and R.S. Gardner, Confidence intervals for the
expectation of a Poisson variable, Biometrika 46 (1959), pp. 441–453.
From the differential distribution of r95% , get all intervals
which contain 68%/95% of possible limits
e.g. on right plot, both ranges
[4.1, 7.7] and [4.6, 9.4]
contain 68% of possible limits
Take the interval corresponding to minimum range
Jan 27, 2011 Mingshui Chen ( University of Florida )
7
An example with different conventions
s = b = 10 / (m/100)^2
no systematics, Bayesian with flat prior
"mean“ (blue solid) and "Asimov" (red solid) are the same for all three plots
With interpolation
Jan 27, 2011 Mingshui Chen ( University of Florida )
Step function
Minimum-range
principle
8
Summary: possible options to choose from
"Typical" expected result :
mean
smooth
(almost no convention)
smooth
Asimov
(will it work for all methods?)
median
Fermi interpolation
smooth
(requires convention, e.g. 50%
quintile)
linear interpolation
smooth
step function
jigged
"Green"/"yellow" bands require convention, e.g.
quantiles
Fermi interpolation
smooth
(2.5%, 16%, 84%, 97.5%)
linear interpolation
smooth
step function
jigged
minimum-range principle
Jan 27, 2011 Mingshui Chen ( University of Florida )
jigged
9
```
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