1. 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 2

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. 3 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 R 2,M R 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. Events R2R2 R2R2 R2R2 R2R2 4 Contents Background Data Setting up Model Single Bin All Bins Jeffreys Prior Results

5. Setting up the Model Each 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. 5 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 here Prior 6 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 ηη Little evidence for dark matter in this bin More evidence for dark matter in this bin 7 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 here Prior 8 Contents 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 here Prior 9 Contents 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 10 62 terms to get a decent estimate… for a single point… We could not find any papers treating a Jeffreys prior for our model either. 10 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: 10 62 terms for an estimate27 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. 11 Contents Background Data Setting up Model Single Bin All Bins Jeffreys Prior Results

12. Here we have the posterior using the Jeffreys Prior, as well as a comparison with using a uniform prior. Blue – Using Jeffreys Prior Green – Using Uniform Prior Results from this Bayesian analysis agreed with a frequentist approach as well. 12 Contents Background Data Setting up Model Single Bin All Bins Jeffreys Prior Results

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