Bayesian Methods in Cosmology

✍ Scribed by Michael P. Hobson, Andrew H. Jaffe, Andrew R. Liddle, Pia Mukherjee, David Parkinson

• Publisher: Cambridge University Press
• Year: 2010
• Tongue: English
• Leaves: 317
• Edition: 1
• Category: Library

✦ Synopsis

In recent years cosmologists have advanced from largely qualitative models of the Universe to precision modelling using Bayesian methods, in order to determine the properties of the Universe to high accuracy. This timely book is the only comprehensive introduction to the use of Bayesian methods in cosmological studies, and is an essential reference for graduate students and researchers in cosmology, astrophysics and applied statistics. The first part of the book focuses on methodology, setting the basic foundations and giving a detailed description of techniques. It covers topics including the estimation of parameters, Bayesian model comparison, and separation of signals. The second part explores a diverse range of applications, from the detection of astronomical sources (including through gravitational waves), to cosmic microwave background analysis and the quantification and classification of galaxy properties. Contributions from 24 highly regarded cosmologists and statisticians make this an authoritative guide to the subject.

✦ Table of Contents

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
List of contributors......Page 11
Preface......Page 13
Part I Methods......Page 15
1.1 Rational inference......Page 17
1.2 Foundations......Page 18
1.2.1 Lattices......Page 19
Assignment......Page 20
Multiplication......Page 21
1.2.3 Information......Page 22
1.2.4 Probability......Page 23
1.3.1 Bayes' theorem......Page 25
1.3.2 Prior probability......Page 26
Symmetry......Page 27
Maximum entropy......Page 28
Continuous problems......Page 29
Geometry......Page 31
1.4 Algorithms......Page 34
Theory......Page 35
Kernel......Page 36
Uncertainty......Page 37
Exploring the prior......Page 38
Posterior......Page 40
Theory......Page 41
Uncertainty......Page 42
Exploration......Page 43
1.4.3 Comparison......Page 44
1.5 Concluding remarks......Page 46
References......Page 48
2.1 Introduction......Page 50
2.2.1 Standard rulers and candles......Page 51
2.2.2 Motivation......Page 54
2.3 Theorists and pre-processed data......Page 55
2.4 Experimentalists and raw measurements......Page 63
2.5 Concluding remarks......Page 68
References......Page 69
3.1 Why do sampling?......Page 71
3.2.1 Direct sampling methods......Page 73
3.2.2 Problems with large dimensions......Page 75
3.2.3 Markov chain sampling......Page 76
3.2.4 Metropolis–Hastings algorithm......Page 77
3.2.5 Other sampling methods......Page 80
3.2.6 Thermodynamic and flat-histogram methods......Page 81
3.2.7 Baby and toy......Page 82
3.3 Have I taken enough samples yet?......Page 83
3.4.1 Parameter constraints......Page 84
3.4.2 Importance sampling......Page 86
3.4.3 Inference from simulation......Page 88
3.4.4 Model selection as parameter estimation......Page 89
3.5 Conclusions......Page 91
References......Page 92
4.1 Introduction......Page 93
4.2 Levels of Bayesian inference......Page 94
4.3 The Bayesian framework......Page 96
4.3.1 Priors......Page 98
4.3.2 Information and complexity......Page 99
4.4.1 General Monte Carlo methods......Page 101
4.4.2 Restricted Monte Carlo methods......Page 102
4.5 Interpretational scales......Page 103
4.6.1 Applications to real data......Page 104
The spectral tilt......Page 105
Dark energy......Page 106
Other applications......Page 109
4.7 Conclusions......Page 110
References......Page 111
5.1 Introduction......Page 113
5.2.1 Utility, expected utility and optimization......Page 114
5.2.2 Choosing the best experiment – an example......Page 115
5.3.1 Fisher matrix error forecast......Page 120
5.3.2 Utility functions for error minimization......Page 122
5.3.3 Application to cosmology: optimization of the WFMOS survey......Page 123
5.4.1 Quantifying experimental capabilities using Bayes factors......Page 129
5.4.2 Application: dark energy vs. a cosmological constant......Page 132
5.5.1 Predictive distributions......Page 134
5.5.3 Application: spectral index from the Planck satellite......Page 136
5.6 Summary......Page 138
References......Page 139
6 Signal separation in cosmology......Page 140
6.1 Model of the data......Page 141
6.2 The hidden, visible and data spaces......Page 142
6.3.1 Mixing matrix......Page 143
6.3.2 Component fields......Page 144
6.4 Choice of data space......Page 147
6.4.1 Pixel-domain data space......Page 149
6.4.2 Fourier-domain data space......Page 150
6.5 Applying Bayes' theorem......Page 151
6.5.1 Defining the posterior distribution......Page 152
6.6 Non-blind signal separation......Page 154
WF posterior......Page 155
Optimal values and error estimates......Page 156
Harmonic-space MEM posterior......Page 157
Relationship between the MEM and WF......Page 158
Determination of the regularization constant......Page 159
Accommodation of spatially varying noise and spectral parameters......Page 160
6.6.3 Mixed-space maximum-entropy method......Page 163
6.7.1 Pixel-domain parameter estimation......Page 165
Uncorrelated signals and noise......Page 166
Correlated signals and noise......Page 168
SMICA data model......Page 169
SMICA likelihood......Page 170
SMICA priors......Page 172
6.7.3 Correlated component analysis (CCA)......Page 173
6.7.4 Determining the optimal number of components......Page 175
References......Page 176
Part II Applications......Page 179
7 Bayesian source extraction......Page 181
7.1 Traditional approaches......Page 182
7.2 The Bayesian approach......Page 184
7.2.1 Discrete sources in a background......Page 185
7.2.2 Bayesian inference......Page 186
7.2.3 Defining the posterior distribution......Page 187
7.3 Variable-source-number models......Page 189
7.5 Single-source models......Page 192
7.5.1 Analytic source extraction and the matched filter......Page 196
7.5.2 Iterative source extraction: local maximization......Page 197
7.5.3 Iterative source extraction: global maximization......Page 199
7.5.4 Simultaneous source extraction......Page 200
7.5.5 Pixel-by-pixel source extraction......Page 201
References......Page 205
8.2 Photometric measurements......Page 207
8.3 Classical flux estimation......Page 210
8.4 The source population......Page 213
8.5 Bayesian flux inference......Page 215
8.6 The faintest sources......Page 218
8.7 Practical flux measurement......Page 223
References......Page 225
9.1 A new spectrum......Page 227
9.2 Gravitational wave data analysis......Page 228
9.2.1 The traditional approach......Page 229
9.3 The Bayesian approach......Page 234
9.3.1 Parameter estimation......Page 236
9.3.2 Search strategies......Page 238
9.3.3 Model selection......Page 239
References......Page 242
10.1 Introduction......Page 243
10.2 The CMB as a hierarchical model......Page 245
10.2.1 CMB data......Page 246
10.2.2 From detectors to maps......Page 247
Destripers......Page 248
Noise marginalization......Page 249
Wiener filters......Page 250
10.2.3 From maps to power spectra......Page 251
10.2.4 From spectra to cosmological parameters......Page 253
10.3 Polarization......Page 254
Power spectra and E/B separation......Page 255
10.4.2 Non-Gaussianity......Page 256
References......Page 257
11.1 Introduction......Page 259
11.2 Galaxy distance indicators......Page 261
11.2.1 The calibration problem......Page 262
11.2.2 The estimation problem......Page 263
11.2.3 Applications of galaxy distance indicators......Page 265
11.3 Multilevel models......Page 266
11.3.1 Adjusting source estimates: shrinkage......Page 267
11.3.2 Poisson point process multilevel models......Page 272
11.4 Future directions......Page 275
References......Page 277
12.1 Discovery space......Page 279
12.2 Average versus maximum likelihood......Page 280
12.3 Priors and Malmquist/Eddington bias......Page 282
12.4 Small samples......Page 284
12.5 Measuring a width in the presence of a contaminating population......Page 286
12.6 Fitting a trend in the presence of outliers......Page 289
12.7 What is the number returned by tests such as…......Page 294
12.8 Summary......Page 295
References......Page 296
13.1 Introduction......Page 297
13.2 Template methods......Page 299
13.3 Bayesian methods and non-colour priors......Page 300
13.4 Training methods and neural networks......Page 301
13.5 Errors on photo-z......Page 303
13.7 Comparison of photo-z codes......Page 304
13.8 The role of spectroscopic datasets......Page 306
13.9 Synergy with cosmological probes......Page 308
13.10 Discussion......Page 310
References......Page 311
Index......Page 313