Bayesian Methods for Astrophysics and Particle Physics

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CONTENTS 5 Bayesian Analysis of Multi–Frequency Sunyaev-Zel’dovich Data118 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 The Arcminute Microkelvin Imager . . . . . . . . . . . . . . . . . 121 5.4 Cluster Modelling through the SZ Effect . . . . . . . . . . . . . . 122 5.4.1 Cluster Gas and Temperature Models . . . . . . . . . . . . 123 5.4.2 SZ Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.2.1 Noise Modelling . . . . . . . . . . . . . . . . . . . 126 5.4.2.2 Point Source Contamination . . . . . . . . . . . . 128 5.4.2.3 SZ Likelihood . . . . . . . . . . . . . . . . . . . . 130 5.4.2.4 Analyzing Multi–Frequency Data . . . . . . . . . 131 5.4.3 Quantifying Cluster Detection . . . . . . . . . . . . . . . . 131 5.5 Application to Simulated SZ Observation . . . . . . . . . . . . . . 133 5.5.1 Analysis and Results . . . . . . . . . . . . . . . . . . . . . 134 5.6 Application to MACS 0647+70 . . . . . . . . . . . . . . . . . . . 140 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6 Bayesian Selection of sign µ within mSUGRA in Global Fits Including WMAP5 Results 144 6.1 Abstact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.3 The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3.1 The Choice of Prior Probability Distribution . . . . . . . . 148 6.3.2 The Likelihood . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.4.1 Model Comparison . . . . . . . . . . . . . . . . . . . . . . 156 6.4.2 Updated Parameter Constraints . . . . . . . . . . . . . . . 158 6.4.3 Consistency Check between Different Constraints . . . . . 164 6.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 167 7 Future Work 169 7.1 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 References 193 ix
List of Figures 1.1 Upper left: Data-sets D1 and D2 drawn from a straight line model (solid line) with slope m = 1 and intercept c = 1 and subject to independent Gaussian noise with root mean square σ1 = σ2 = 0.1. Upper right: Posterior P(m, c|D, H1) assuming that data-sets D1 and D2 are consistent. Lower left: Posterior P(m, c|D, H1) for data-set D1. Lower right: Posterior P(m, c|D, H1) for data-set D2. The inner and outer contours enclose 68% and 95% of the total probability respectively. The true parameter value is indicated by red crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Upper left: Data-sets D1 and D2 drawn from a straight line model (solid line) with slope m = 0, c = 1.5 and m = 1, c = 1 respectively and subject to independent Gaussian noise with root mean square σ1 = σ2 = 0.1. Upper right: Posterior P(m, c|D, H1) assuming that data-sets D1 and D2 are consistent. Lower left: Posterior P(m, c|D, H1) for data-set D1. Lower right: Posterior P(m, c|D, H1) for data-set D2. The inner and outer contours enclose 68% and 95% of the total probability respectively. The true parameter values are indicated by red and black crosses for Data- sets D1 and D2 respectively. . . . . . . . . . . . . . . . . . . . . . 10 1.3 Proper thermodynamic integration requires the log–likelihood to be concave like (a), not (b). . . . . . . . . . . . . . . . . . . . . . 16 2.1 Cartoon illustrating (a) the posterior of a two dimensional prob- lem; and (b) the transformed L(X) function where the prior volumes Xi are associated with each likelihood Li. . . . . . . . . . . . . . . 21 x
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 and 12: Contents 1 Bayesian Inference 1 1.1
Page 13: CONTENTS 3.3.5 Parallelization . .
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
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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 Guth, A.H. (1981). Infla
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REFERENCES Jenkins, A., Frenk, C.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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