Bayesian Dark Matter Limits for 8 TeV p-p Collisions at the LHC

Cedric Flamant

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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

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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

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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

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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

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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

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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

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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

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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

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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

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