bayesian dark matter limits for 8 tev p-p collisions at the lhc cedric flamant
TRANSCRIPT
Bayesian Dark Matter Limits for 8 TeV p-p Collisions at the LHC
Cedric Flamant
2
Summary• Background Recap• Where the Data Comes From• Setting up the Model• Likelihood of Data for One Bin• Likelihood of Data in All Bins• Jeffreys Prior Computation• Results
3
Background Recap• Goal: to obtain dark matter signal strength limits using
Bayesian analysis of CMS data.• Data comes from complicated analysis of actual CMS
detections at the LHC.• It has gone through a lot of processing before we conduct
our Bayesian analysis on it.
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
4
The Data• 18.5 fb-1 of raw data analyzed.• Counts are broken up into
rectangles in R2,MR Razor variable space.• Each rectangle has a predicted
Standard Model background shown in green, with error bars.• Data is shown in black, along
with error bars.• Discrepancies could be a sign
of dark matter, the focus of this project.
Even
ts
Even
ts
Even
ts
Even
ts
R2 R2
R2 R2
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
5
Setting up the ModelEach Razor variable bin has an expected number of counts, assuming the existence of dark matter, given by
where b is the standard model background counts, s is the dark matter signal, and η is the signal strength.
We want to get a posterior of signal strength η so we can find the most likely value given the data, and the 95% confidence upper limit.
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
6
Likelihood of Getting the Data in One Bin• The simplest place to start is to find the DM signal strength when only
looking at a single bin.
Where
The integrals are for marginalizing over the systematic errors in s and b that we don’t care to know. We cannot analytically integrate, so we can either use numerical or MCMC methods here.
Observed data comes in herePrior
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
7
Likelihood of Getting the Data in One Bin
• Plotting the above function for different bins results in different most likely values for DM signal strength
Bin 0 Bin 18
η ηLittle evidence for dark matter in this bin More evidence for dark matter in this bin
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
8
Likelihood of Getting the Data in All Bins• We want to consider the entire space for our likelihood:
Where
This case is far trickier than only considering one bin at a time, since numerical integration of this expression is incredibly slow. Thus, we turn to MCMC methods.
Observed data comes in herePriorContents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
9
Likelihood of Getting the Data in All Bins• We want to consider the entire space for our likelihood:
Now comes the question of what prior to use. We could use a uniform prior (which in this case would be an improper prior due to the infinite extent of η), but it tends to bias towards larger values of signal strength.
We need a suitable non-informative prior – How about Jeffreys Prior
Observed data comes in herePriorContents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
10
Jeffreys Prior for Model• It turns out to pretty much look like death
We kind of get stuck here – it would take 1062 terms to get a decent estimate… for a single point…We could not find any papers treating a Jeffreys prior for our model either.
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
11
But, Incredibly -• Kneading the equation for two days, I analytically simplified the
expression to an absurdly simple result:
1062 terms for an estimate 27 terms for an exact result
(math is in appendix)
Simplification confirmed for Nbins = 1 since it’s a known result, and verified to machine precision numerically for Nbins = 2
An Nbins = 3 verification would take weeks.
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
12
Results• Here we have the posterior using the Jeffreys Prior, as well as a
comparison with using a uniform prior.
Blue – Using Jeffreys PriorGreen – Using Uniform Prior
Results from this Bayesian analysis agreed with a frequentist approach as well.
Contents
Background
Data
Setting up Model
Single Bin
All Bins
Jeffreys Prior
Results
13
14
15