1. Exploring SUSY Parameter Space A New Bayesian Approach
J. Lykken3, M. Pierini2, H.B. Prosper4, C. Rogan2, S. Sekmen4, M. Spiropulu1,2
1Caltech, 2CERN, 3FNAL, 4FSU
Characterization of New Physics at the LHC, II

2. Outline
Statement of Problem
General Strategy
An Example
Summary

3. Statement of Problem
“the most pressing question at the LHC will be to figure out whether there is any evidence for physics beyond the standard model, and then most broadly what theoretical framework best describes the new physics” Supersymmetry and the LHC Inverse Problem, N. Arkani-Hamed, G.L.Kane, J. Thaler, L.Wang, JHEP 0608, 070 (2006)

4. Statement of Problem Data
Basic problem: All interesting theories are multi-parameter models SM
me, mμ, mτ
mu, md, ms, mc, mb, mt
θ12, θ23, θ13, δ
g1, g2, g3
θQCD
μ, λ
Data
MSSM
Theory of the Week
Basic questions:
Which theories are preferred, given the data?
And which parameter sub-spaces?

5. Statement of Problem  ★ ✚
Model degeneracy: The map from LHC signatures to parameter sub-spaces
is not one-to-one.
LHC Signatures
MSSM
Theory of the Week ★ ✚ 
Therefore, the most likely outcome of a discovery and its initial
characterization will be a set of degenerate models.
A model = a parameter sub-space

6. Some Remarks
It is not possible to make progress without making some assumption about the nature of potential signals. For example, to model backgrounds using data one typically assumes that: signal << background in some control region. But, a priori, we do know whether the control region we have chosen enjoys this property. Therefore, as was the case with the characterization of the SM, the inverse problem will be solved iteratively.

7. General Strategy

8. General Strategy What do we wish to do…over the next decade or so?
Answer the question: is there any evidence of new physics?
Rank any collection of new physics models.
Characterize the parameters of all viable models.
Iteratively converge to the most viable new theory of physics.
We propose using Bayesian methods to guide this research program.
But this requires imposing probability densities p(θ) on parameter spaces.

9. General Strategy
p(θ) is controversial (witness JE). But, p(θ) is also extremely useful. Example: Suppose we wish to distinguish the xMSSM from the SM. After a huge amount of work, the SM has been confined to a small sub-space of the SM parameter space, the blue spot. This is not the case for the xMSSM. Suppose, you have p(θ) for the xMSSM, then from a Bayesian perspective the way forward is clear: maximize the “separation” between the blue spot and the green blob, using p(θ) as a weighting function. SM
xMSSM

10. Bayes in 60 Seconds
Given a model with parameters θ, in principle, any inverse problem can solved using Bayes theorem:
Posterior density of model given x
Likelihood
of data x
Prior density of model
Appealing features:
General and conceptually straightforward
Systematic learning from data through a recursive algorithm
Coherent way to incorporate uncertainties regardless of their origin
Posterior density is the complete inference for a given model
Can rank models according to their concordance with observations
Principal difficulty: construction of priors

11. The Issue with Priors
The construction of p(θ) is controversial because answers depend on it. Example: Lopez-Fogliani et al. arXiv: v1 Left plot based on prior flat in m0, m1/2 Right plot based on prior flat in log m0, m1/2 However, the issue is not that one gets different answers. Rather the issue is in how the priors are chosen.

12. Reference Priors
In 1979, J. Bernardo introduced a method for constructing priors specifically designed to minimize their influence relative to the likelihood. He called these reference priors. By definition, a reference prior π(θ) maximizes the separation between it and the posterior density. The separation measure used is the Kullback-Leibler divergence between the posterior density p(θ|x) and the prior π(θ). In practice, one uses D[π, p] averaged over all possible data-sets from K replications of the experiment, and let K go to infinity.

13. Reference Priors
If the posterior density p(θ | x) is asymptotically normal, the reference prior π(θ) for models with one continuous parameter reduces to the Jeffreys prior (the square-root of the Fisher information F): This simplifies calculations considerably. The reference prior construction can be extended to more than one parameter; however, the single-parameter algorithm is sufficient for our purposes.

14. An Example

15. The xMSSM Alas poor SUSY sapiens…I knew them well… WARNING
SUSY PARAMETER SPACE
Alas poor SUSY sapiens…I knew them well…

16. Is there Evidence of a Signal?
Construct a likelihood p(n|s, μ)
n = observed count
s = expected signal
μ = expected background
Construct a prior π(s, μ) = π(s| μ) π(μ)
Use a reference prior for π(s|μ)*
Use, for example, a gamma prior for π(μ)
Compute the posterior density p(s, μ|n) ~ p(n|s, μ) π(s, μ)
* “Reference priors for high energy physics,”
L. Demortier, S. Jain, and H.B. Prosper, Phys.Rev.D82:034002, 2010

17. Is there Evidence of a Signal?
Given the posterior density p(s, μ|n) and a measure of the separation between the s > 0 models and the s = 0 model, this question can be answered. We again use the Kullback-Leibler divergence, KL, to quantify the separation between models. But, since we know neither s nor μ, it is necessary to average over all possible values of s and μ: where KL(s,μ) = -s + (s+μ) ln(1+s/μ) is the KL divergence of a given s > 0 model from the s = 0 model. We call B(n) the Bayes signal significance.

18. Initial Characterization
From the posterior density p(s, μ|n), we can compute which encapsulates what we know about the signal, given the count n, Since s = f(θ), p(s|n) also contains information about the xMSSM: To proceed further, we make the simplest possible assumption: points that yield the same signal are equi-probable.
xMSSM δ Δ

19. Example – CMSSM
We illustrate this approach with a simple class of models (CMSSM):
free parameters: 150 < m0 < 600 and 0 < m1/2 < 1500
fixed parameters: A0 = 0, tanβ = 10 and μ > 0
We use the CMS SUSYbenchmark point LM1 with
m0 = 60, m1/2 = 250, A0 = 0, tanβ = 10, μ > 0
as the “true state of nature”, which will provide the observed count n.
For LM1 and for each point in a grid in the m0-m1/2 space, we generate TeV LHC events (using PYTHIA and PGS), do a simple analysis, and quote results for 1pb-1, 100 pb-1 and 500 pb-1.

20. Example – CMSSW
xMSSM δ Δ

21. Example – CMSSW
Now include EW/flavor observables (BR(b -> sγ), R(BR(b -> τν)), BR(b -> Dτν), BR(b -> Dτν)/BR(b -> eτν), Rl23, BR(Ds -> τν), BR(Ds -> μν) and Δρ).
Since the the state of nature is LM1, we use the LM1 values for the observables along with the measured uncertainties in this example.

22. Summary
Our goal over the next decade or so is to converge to the NSM (the New Standard Model).
To do so, we need a well-defined way to rank models.
This requires assigning a number to each model that permits comparison. The peak of the likelihood function cannot do this and likelihood ratios are only a partial answer.
A Bayesian strategy using reference analysis, of which the reference prior is the key element, provides a well-founded way forward.
We are currently working on its application to realistic models.