Bayesian Methods for Astrophysics and Particle Physics

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LIST OF FIGURES 3.6 1–D Marginalized posteriors for flat-ΛCDM Cosmology with Cos- moMC and 48,000 likelihood evaluations (solid black), CosmoMC and 3,200 likelihood evaluations (dotted pink) and MultiNest with 3,100 likelihood evaluations (dashed blue). . . . . . . . . . . 89 4.1 Samples drawn from the Press–Schechter mass function. The con- tour enclose 68%, 95% and 99% of the probability. . . . . . . . . . 100 4.2 Left panel shows the true convergence map of the simulated clusters discussed in Section 4.4.1. Middle panel shows the denoised convergence map made with the LenEnt2 algorithm. Right panel shows the inferred convergence map made with the MultiNest algorithm.102 4.3 The expected and actual number of false positives as function of the “threshold probability” pth (see Section 4.3.3). . . . . . . . . . 104 4.4 Purity and completeness as a function of the “threshold probability” for the analysis of the simulated weak lensing data described in Section 4.4.2. Press–Schechter mass function was used as a joint prior on M200 and z.. . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5 Receiver Operating Characteristic (ROC) curve obtained by vary- ing the threshold probability pth used in identifying candidate clusters for the analysis of simulated weak lensing data–set discussed in Section 4.4.2. The points are labelled with their corres- ponding pth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.6 Parameters of the clusters detected by the MultiNest technique applied to the simulation discussed in Section 4.4.1. Red (solid) bars correspond to the cluster for which R > 1 and green (dashed) bars to the ones with R < 0. All error–bars are 1–σ error–bars. . . 106 4.7 Inferred parameters against true parameters for true cluster de- tections. Green crosses represent the true cluster parameters and the red bars the inferred parameter values. All error–bars are 1–σ error–bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 xiii
LIST OF FIGURES 4.8 The total number of clusters in a given redshift and mass bin in the true catalogue (red) against the number of detected clusters (green) for the analysis with weak lensing simulation discussed in Section 4.5.1 with pth = 0.5. . . . . . . . . . . . . . . . . . . . . . 110 4.9 Completeness (red) and purity (green) for the analysis with weak lensing simulation discussed in Section 4.5.1 with pth = 0.5. Press– Schechter mass function was used as a joint prior on M200 and z. . 111 4.10 Completeness in mass-redshift plane for the analysis with weak lensing simulation discussed in Section 4.5.1 with pth = 0.5. Press– Schechter mass function was used as a joint prior on M200 and z. . 111 4.11 Purity and completeness as a function of the “threshold probability” for the analysis of the simulated weak lensing data described in Section 4.5.2. Press–Schechter mass function was used as a joint prior on M200 and z. . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.12 Inferred parameter values (on the y–axis) against the true parameter values (on the x–axis) from the cluster catalogue for each of the detected clusters. The analysis was done with pth = 0.5 using (a) Press-Schechter mass function prior on M200 and z and (b) logarithmic prior on M200 and uniform prior on z. . . . . . . . 113 4.13 Difference between the inferred parameter values and the true parameter values from the cluster catalogue, plotted for each detected clusters. The analysis was done with pth = 0.5 using (a) Press-Schechter mass function prior on M200 and z and (b) logarithmic prior on M200 and uniform prior on z. . . . . . . . . . . . . 114 4.14 Difference between the inferred parameter values and the true parameter values from the cluster catalogue, plotted against the true parameter values for each of the detected clusters. The analysis was done with pth = 0.5 using (a) Press-Schechter mass function prior on M200 and z and (b) logarithmic prior on M200 and uniform prior on z. . . . . . . . . . . . . . . . . . . . . . . . . . . 115 xiv
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
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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 for Astronomy II. Edited
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REFERENCES Stark, L.S., Hafliger, P
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