Bayesian Methods in Cosmology

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.

Michael P. Hobson is Reader in Astrophysics and Cosmology at the Cavendish Laboratory, University of Cambridge, where he researches theoretical and observational cosmology, Bayesian statistical methods, gravitation and theoretical optics. Andrew H. Jaffe is Professor of Astrophysics and Cosmology at Imperial College, London, and a member of the Planck Surveyor Satellite collaboration, which will create the highest-resolution and most sensitive maps of the cosmic microwave background ever produced. Andrew R. Liddle is Professor of Astrophysics at the University of Sussex. He is the author of over 150 journal articles and four books on cosmology, covering topics from early universe theory to modelling astrophysical data. Pia Mukherjee is a Postdoctoral Research Fellow in the Astronomy Centre at the University of Sussex, specializing in constraining cosmological models, including dark energy models, from observational data. David Parkinson is a Postdoctoral Research Fellow in the Astronomy Centre at the University of Sussex, working in the areas of cosmology and the early Universe.

List of contributors	p. ix
Preface	p. xi
Methods	p. 1
Foundations and algorithms	p. 3
Rational inference	p. 3
Foundations	p. 4
Inference	p. 11
Algorithms	p. 20
Concluding remarks	p. 32
Simple applications of Bayesian methods	p. 36
Introduction	p. 36
Essentials of modern cosmology	p. 37
Theorists and pre-processed data	p. 41
Experimentalists and raw measurements	p. 49
Concluding remarks	p. 54
Parameter estimation using Monte Carlo sampling	p. 57
Why do sampling?	p. 57
How do I get the samples?	p. 59
Have I taken enough samples yet?	p. 69
What do I do with the samples?	p. 70
Conclusions	p. 77
Model selection and multi-model inference	p. 79
Introduction	p. 79
Levels of Bayesian inference	p. 80
The Bayesian framework	p. 82
Computing the Bayesian evidence	p. 87
Interpretational scales	p. 89
Applications	p. 90
Conclusions	p. 96
Bayesian experimental design and model selection forecasting	p. 99
Introduction	p. 99
Predicting the effectiveness of future experiments	p. 100
Experiment optimization for error reduction	p. 106
Experiment optimization for model selection	p. 115
Predicting the outcome of model selection	p. 120
Summary	p. 124
Signal separation in cosmology	p. 126
Model of the data	p. 127
The hidden, visible and data spaces	p. 128
Parameterization of the hidden space	p. 129
Choice of data space	p. 133
Applying Bayes' theorem	p. 137
Non-blind signal separation	p. 140
(Semi-)blind signal separation	p. 151
Applications	p. 165
Bayesian source extraction	p. 167
Traditional approaches	p. 168
The Bayesian approach	p. 170
Variable-source-number models	p. 175
Fixed-source-number models	p. 178
Single-source models	p. 178
Conclusions	p. 191
Flux measurement	p. 193
Introduction	p. 193
Photometric measurements	p. 193
Classical flux estimation	p. 196
The source population	p. 199
Bayesian flux inference	p. 201
The faintest sources	p. 204
Practical flux measurement	p. 209
Gravitational wave astronomy	p. 213
A new spectrum	p. 213
Gravitational wave data analysis	p. 214
The Bayesian approach	p. 220
Bayesian analysis of cosmic microwave background data	p. 229
Introduction	p. 229
The CMB as a hierarchical model	p. 231
Polarization	p. 240
Complications	p. 242
Conclusions	p. 243
Bayesian multilevel modelling of cosmological populations	p. 245
Introduction	p. 245
Galaxy distance indicators	p. 247
Multilevel models	p. 252
Future directions	p. 261
A Bayesian approach to galaxy evolution studies	p. 265
Discovery space	p. 265
Average versus maximum likelihood	p. 266
Priors and Malmquist/Eddington bias	p. 268
Small samples	p. 270
Measuring a width in the presence of a contaminating population	p. 272
Fitting a trend in the presence of outliers	p. 275
What is the number returned by tests such as x², KS, etc.?	p. 280
Summary	p. 281
Photometric redshift estimation: methods and applications	p. 283
Introduction	p. 283
Template methods	p. 285
Bayesian methods and non-colour priors	p. 286
Training methods and neural networks	p. 287
Errors on photo-z	p. 289
Optimal filters	p. 290
Comparison of photo-z codes	p. 290
The role of spectroscopic datasets	p. 292
Synergy with cosmological probes	p. 294
Discussion	p. 296
Index	p. 299
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Bayesian Methods in Cosmology

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