Bayesian Methods for Astrophysics and Particle Physics

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CONTENTS 2.4.2 Recursive Clustering . . . . . . . . . . . . . . . . . . . . . 27 2.5 Improved Ellipsoidal Sampling Methods . . . . . . . . . . . . . . 28 2.5.1 General Improvements . . . . . . . . . . . . . . . . . . . . 28 2.5.1.1 Identification of Clusters . . . . . . . . . . . . . . 28 2.5.1.2 Dynamic Enlargement Factor . . . . . . . . . . . 29 2.5.1.3 Detection of Overlapping Ellipsoids . . . . . . . . 30 2.5.1.4 Sampling from Overlapping Ellipsoids . . . . . . 30 2.5.2 Method 1: Simultaneous Ellipsoidal Sampling . . . . . . . 31 2.5.3 Method 2: Clustered Ellipsoidal Sampling . . . . . . . . . 32 2.5.4 Evaluating ‘Local’ Evidences . . . . . . . . . . . . . . . . . 33 2.5.4.1 Method 1 . . . . . . . . . . . . . . . . . . . . . . 34 2.5.4.2 Method 2 . . . . . . . . . . . . . . . . . . . . . . 35 2.5.5 Dealing with Degeneracies . . . . . . . . . . . . . . . . . . 36 2.6 Metropolis Nested Sampling . . . . . . . . . . . . . . . . . . . . . 38 2.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.1 Toy Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.2 Toy Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Bayesian Object Detection . . . . . . . . . . . . . . . . . . . . . . 49 2.8.1 Discrete Objects in Background . . . . . . . . . . . . . . . 50 2.8.2 Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . 51 2.8.3 Defining the Posterior Distribution . . . . . . . . . . . . . 52 2.8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.9 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . 57 3 MultiNest: An Efficient & Robust Bayesian Inference Tool for Cosmology & Particle Physics 61 3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 The MultiNest Algorithm . . . . . . . . . . . . . . . . . . . . . 62 3.3.1 Unit Hypercube Sampling Space . . . . . . . . . . . . . . . 63 3.3.2 Partitioning and Construction of Ellipsoidal Bounds . . . . 64 3.3.3 Sampling from Overlapping Ellipsoids . . . . . . . . . . . . 70 3.3.4 Decreasing the Number of Active Points . . . . . . . . . . 70 vii
CONTENTS 3.3.5 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.6 Identification of Modes . . . . . . . . . . . . . . . . . . . . 72 3.3.7 Evaluating ‘Local’ Evidences . . . . . . . . . . . . . . . . . 75 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4.1 Toy Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4.2 Toy Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5 Cosmological Model Selection . . . . . . . . . . . . . . . . . . . . 82 3.5.1 Results: The Spatial Curvature . . . . . . . . . . . . . . . 84 3.5.2 Results: Varying Equation of State of Dark Energy . . . . 85 3.6 Comparison of MultiNest and MCMC ‘quick look’ Parameter Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . 88 4 Cluster Detection in Weak Lensing Surveys 93 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.1 Weak Lensing Background . . . . . . . . . . . . . . . . . . 95 4.3.2 Cluster Mass Models . . . . . . . . . . . . . . . . . . . . . 97 4.3.3 Quantifying Cluster Detection . . . . . . . . . . . . . . . . 98 4.4 Application to Mock Data: Recovery of Toy Model Parameters . . . . . . . . . . . . . . . . . 101 4.4.1 A Simple Simulated Data–Set . . . . . . . . . . . . . . . . 101 4.4.2 Analysis and Results . . . . . . . . . . . . . . . . . . . . . 101 4.5 Application to Mock Data: Wide–Field Numerical Simulations . . . . . . . . . . . . . . . . . 105 4.5.1 Mock Shear Survey Data . . . . . . . . . . . . . . . . . . . 108 4.5.2 Analysis and Results . . . . . . . . . . . . . . . . . . . . . 108 4.5.2.1 Completeness and Purity . . . . . . . . . . . . . 109 4.5.2.2 Accuracy of Parameter Estimates . . . . . . . . . 112 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 viii
Page 1: Bayesian Methods for Astrophysics a
Page 5: I would like to dedicate this thesi
Page 9 and 10: Abstract This thesis is concerned w
Page 11: Contents 1 Bayesian Inference 1 1.1
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
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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
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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 Allanach, B.C., Cranmer,
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REFERENCES Beltrán, M., García-Be
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REFERENCES Condon, J.J. (1974). Con
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REFERENCES Ellis, J., Heinemeyer, S
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REFERENCES Li, C.T. et al. (2006).
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REFERENCES Misiak, M. & Steinhauser
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REFERENCES for Astronomy II. Edited
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REFERENCES Stark, L.S., Hafliger, P
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REFERENCES Waldram, E.M., Bolton, R
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