Bayesian Methods for Astrophysics and Particle Physics

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The impending start-up of the Large Hadron Collider (LHC) makes this a very exciting time for the supersymmetric (SUSY) phenomeno- logy especially for analyzing different models of SUSY breaking at low energies e.g. mSUGRA, SplitSUSY, non-universal mSUGRA, AMSB, etc. Most of these models do not have a simple way of confirmation and some statistical model comparison will be required on the data coming from the LHC. We apply the MultiNest technique for the analysis of mSUGRA model to distinguish between its two branches (concerned with the sign of µ–the Higgs/higgsino mass parameter). We conclude that the current data has a moderate preferance for µ > 0.
Contents 1 Bayesian Inference 1 1.1 Bayesian Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Marginalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Bayesian Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Combining Data–Sets . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Comparing Data–Sets . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.1 Toy Problem . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5.1.1 Case I: Consistent Data-Sets . . . . . . . . . . . 7 1.5.1.2 Case II: Inconsistent Data-Sets . . . . . . . . . . 9 1.6 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.1 Markov Chain Monte Carlo (MCMC) Sampling . . . . . . 11 1.6.1.1 Problems with Metropolis–Hastings Algorithm . . 13 1.6.2 Thermodynamic Integration . . . . . . . . . . . . . . . . . 14 1.6.2.1 Problems with Thermodynamic Integration . . . 15 2 Multimodal Nested Sampling 17 2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Nested Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Evidence Evaluation . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Posterior Inferences . . . . . . . . . . . . . . . . . . . . . . 23 2.3.4 Evidence Error Estimation . . . . . . . . . . . . . . . . . . 23 2.4 Ellipsoidal Nested Sampling . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Single Ellipsoid Sampling . . . . . . . . . . . . . . . . . . . 26 vi
Page 1: Bayesian Methods for Astrophysics a
Page 5: I would like to dedicate this thesi
Page 9: Abstract This thesis is concerned w
Page 13 and 14: CONTENTS 3.3.5 Parallelization . .
Page 15 and 16: List of Figures 1.1 Upper left: Dat
Page 17 and 18: LIST OF FIGURES 2.9 The toy model d
Page 19 and 20: LIST OF FIGURES 4.8 The total numbe
Page 21: LIST OF FIGURES 6.2 The 2-dimension
Page 24 and 25: 1.1 Bayesian Modelling Two fundamen
Page 26 and 27: |∆ log E| Odds Probability Remark
Page 28 and 29: 1.5 Comparing Data-Sets This joint
Page 30 and 31: c 1.6 1.4 1.2 1 0.8 0.6 (a) 0.4 0.6
Page 32 and 33: c 2 1.5 1 (a) 0.5 −1 0 1 2 m (c)
Page 34 and 35: of the (i + 1) th point is given as
Page 36 and 37: 1.6 Numerical Methods posterior dis
Page 38 and 39: log L 1 Anneal log X (a) 0 F E D 1.
Page 40 and 41: 2.2 Introduction modes and signific
Page 42 and 43: 2.3 Nested Sampling third algorithm
Page 44 and 45: 2.3 Nested Sampling Nested sampling
Page 46 and 47: (a) (b) (c) (d) (e) 2.3 Nested Samp
Page 48 and 49: 2.4 Ellipsoidal Nested Sampling H/
Page 50 and 51: 2.5 Improved Ellipsoidal Sampling M
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2.6 Metropolis Nested Sampling volu
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2.7 Applications iteration since th
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2.7 Applications Figure 2.6: As in
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Peak X Y Local log Z 1 −0.400 ±
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L 5 4 3 2 1 0 -5 -2.5 0 x 2.5 5 -5
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48 Analytical Method 2 (with sub-cl
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2.8 Bayesian Object Detection Metho
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Object X Y A R 1 43.71 22.91 10.54
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Log-Likelihood (L) -84800 -84900 10
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2.8 Bayesian Object Detection Metho
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2.9 Discussion and Conclusions lar
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2.9 Discussion and Conclusions elli
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3.2 Introduction 3.2 Introduction I
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3.3 The MultiNest Algorithm The phy
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where γ is a constant, 3.3 The Mul
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(a) (b) 3.3 The MultiNest Algorithm
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3.3 The MultiNest Algorithm greater
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3.3.6 Identification of Modes 3.3 T
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1 u 2 G 5 G 6 G 3 G 7 3.3 The Multi
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3.3 The MultiNest Algorithm For eac
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250 200 150 100 50 0 0 250 200 150
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Mode true local log(Z) MultiNest lo
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3.5 Cosmological Model Selection fr
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0.018 ≤ Ωbh 2 ≤ 0.032 0.04
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3.6 Comparison of MultiNest and MCM
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3.7 Discussion and Conclusions dete
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3.7 Discussion and Conclusions As a
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3.7 Discussion and Conclusions For
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4.2 Introduction The number count o
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4.3 Methodology The unlensed ellipt
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4.3.3 Quantifying Cluster Detection
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4.3 Methodology Figure 4.1: Samples
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4.4 Application to Mock Data: Recov
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False Positives 20 18 16 14 12 10 8
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c log R 16 14 12 10 8 6 4 2 0 0 0.5
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4.5.1 Mock Shear Survey Data 4.5 Ap
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No. of Clusters 1000 100 10 1 0 0.2
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1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0
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∆x/arcsec ∆z ∆x/arcsec ∆z 4
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4.6 Conclusions the true parameters
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Chapter 5 Bayesian Analysis of Mult
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5.2 Introduction intensive Markov C
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5.4 Cluster Modelling through the S
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5.4.2 SZ Data 5.4 Cluster Modelling
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p(α) 1 0.8 0.6 0.4 0.2 0 5.4 Clust
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5.5 Application to Simulated SZ Obs
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5.5 Application to Simulated SZ Obs
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T/keV 20 18 16 14 12 10 8 6 4 2 M /
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5.6 Application to MACS 0647+70 Par
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y 0 /arcsec r core /h −1 kpc β M
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Chapter 6 Bayesian Selection of sig
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6.2 Introduction performing a multi
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mSUGRA parameters 2 TeV range 4 TeV
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6.3 The Analysis Observable Mean va
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6.3 The Analysis The W boson pole m
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6.3 The Analysis The SM prediction
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6.4 Results 6.4 Results In this sec
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m0 (TeV) 4 3.5 3 2.5 2 1.5 1 0.5 0.
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m0 (TeV) 4 3.5 3 2.5 2 1.5 1 0.5 0.
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P/Pmax P/Pmax 1 0.75 0.5 0.25 0 1 0
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6.4 Results as shown previously by
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m0 (TeV) m0 (TeV) m0 (TeV) m0 (TeV)
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6.5 Summary and Conclusions predict
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7.1 Analysis Methods provides a ver
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REFERENCES Stark, L.S., Hafliger, P
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REFERENCES Waldram, E.M., Bolton, R
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