Contents

PART I THE ESSENTIALS

1. The basics 3

1.1 Introduction: deductive logic versus plausible reasoning 3

1.2 Probability: Cox and the rules for consistent reasoning 4

1.3 Corollaries: Bayes’ theorem and marginalization 5

1.4 Some history: Bayes, Laplace and orthodox statistics 8

1.5 Outline of book 12

2. Parameter estimation I 14

2.1 Example 1: is this a fair coin? 14

2.1.1 Different priors 17

2.1.2 Sequential or one-step data analysis? 19

2.2 Reliabilities: best estimates, error-bars and confidence intervals 20

2.2.1 The coin example 23

2.2.2 Asymmetric posterior pdfs 24

2.2.3 Multimodal posterior pdfs 25

2.3 Example 2: Gaussian noise and averages 26

2.3.1 Data with different-sized error-bars 29

2.4 Example 3: the lighthouse problem 29

2.4.1 The central limit theorem 33

3. Parameter estimation II 35

3.1 Example 4: amplitude of a signal in the presence of background 35

3.1.1 Marginal distributions 39

3.1.2 Binning the data 42

3.2 Reliabilities: best estimates, correlations and error-bars 43

3.2.1 Generalization of the quadratic approximation 49

3.2.2 Asymmetric and multimodal posterior pdfs 50

3.3 Example 5: Gaussian noise revisited 52

3.3.1 The Student-t and χ2 distributions 54

3.4 Algorithms: a numerical interlude 55

3.4.1 Brute force and ignorance 56

3.4.2 The joys of linearity 57

3.4.3 Iterative linearization 58

3.4.4 Hard problems 60

3.5 Approximations: maximum likelihood and least-squares 61

3.6 Error-propagation: changing variables 68

3.6.1 A useful short cut 73

3.6.2 Taking the square root of a number 74

4. Model selection 78

4.1 Introduction: the story of Mr A and Mr B 78

4.1.1 Comparison with parameter estimation 83

4.1.2 Hypothesis testing 84

4.2 Example 6: how many lines are there? 85

4.2.1 An algorithm 89

4.2.2 Simulated data 91

4.2.3 Real data 93

4.3 Other examples: means, variance, dating and so on 94

4.3.1 The analysis of means and variance 94

4.3.2 Luminescence dating 98

4.3.3 Interlude: what not to compute 100

5. Assigning probabilities 103

5.1 Ignorance: indifference and transformation groups 103

5.1.1 The binomial distribution 107

5.1.2 Location and scale parameters 108

5.2 Testable information: the principle of maximum entropy 110

5.2.1 The monkey argument 113

5.2.2 The Lebesgue measure 115

5.3 MaxEnt examples: some common pdfs 117

5.3.1 Averages and exponentials 117

5.3.2 Variance and the Gaussian distribution 118

5.3.3 MaxEnt and the binomial distribution 120

5.3.4 Counting and Poisson statistics 121

5.4 Approximations: interconnections and simplifications 121

5.5 Hangups: priors versus likelihoods 124

5.5.1 Improper pdfs 124

PART II ADVANCED TOPICS

6. Non-parametric estimation 129

6.1 Introduction: free-form solutions 129

6.1.1 Singular value decomposition 130

6.1.2 A parametric free-form solution? 135

6.2 MaxEnt: images, monkeys and a non-uniform prior 136

6.2.1 Regularization 138

6.3 Smoothness: fuzzy pixels and spatial correlations 140

6.3.1 Interpolation 141

6.4.1 Summary of the basic strategy 144

6.4.2 Inference or inversion? 145

6.4.3 Advanced examples 148

7. Experimental design 149

7.1 Introduction: general issues 149

7.2 Example 7: optimizing resolution functions 151

7.2.1 An isolated sharp peak 152

7.2.2 A free-form solution 156

7.3 Calibration, model selection and binning 161

7.4 Information gain: quantifying the worth of an experiment 163

8. Least-squares extensions 165

8.1 Introduction: constraints and restraints 165

8.2 Noise scaling: a simple global adjustment 166

8.3 Outliers: dealing with erratic data 167

8.3.1 A conservative formulation 168

8.3.2 The good-and-bad data model 171

8.3.3 The Cauchy formulation 172

8.4 Background removal 173

8.5 Correlated noise: avoiding over-counting 174

8.5.1 Nearest-neighbour correlations 175

8.5.2 An elementary example 176

8.5.3 Time series 177

8.6 Log-normal: least-squares for magnitude data 179

9. Nested sampling 181

9.1 Introduction: the computational problem 181

9.1.1 Evidence and posterior 182

9.2 Nested sampling: the basic idea 184

9.2.1 Iterating a sequence of objects 185

9.2.2 Terminating the iterations 186

9.2.3 Numerical uncertainty of computed results 187

9.2.4 Programming nested sampling in ‘C’ 188

9.3 Generating a new object by random sampling 190

9.3.1 Markov chain Monte Carlo (MCMC) exploration 191

9.3.2 Programming the lighthouse problem in ‘C’ 192

9.4 Monte Carlo sampling of the posterior 195

9.4.1 Posterior distribution 196

9.4.2 Equally-weighted posterior samples: staircase sampling 197

9.4.3 The lighthouse posterior 198

9.4.4 Metropolis exploration of the posterior 199

9.5 How many objects are needed? 200

9.5.1 Bi-modal likelihood with a single ‘gate’ 200

9.6 Simulated annealing 203

9.6.1 The problem of phase changes 203

9.6.2 Example: order/disorder in a pseudo-crystal 204

9.6.3 Programming the pseudo-crystal in ‘C’ 206

10. Quantification 209

10.1 Exploring an intrinsically non-uniform prior 209

10.1.1 Binary trees for controlling MCMC transitions 210

10.2 Example: ON/OFF switching 212

10.2.1 The master engine: flipping switches individually 212

10.2.2 Programming which components are present 212

10.2.3 Another engine: exchanging neighbouring switches 215

10.2.4 The control of multiple engines 216

10.3 Estimating quantities 216

10.3.1 Programming the estimation of quantities in ‘C’ 218

10.4 Final remarks 223

A. Gaussian integrals 224

A.1 The univariate case 224

A.2 The bivariate extension 225

A.3 The multivariate generalization 226

B. Cox’s derivation of probability 229

B.1 Lemma 1: associativity equation 232

B.2 Lemma 2: negation 235

Bibliography 237