Table of Preface page xi PART I INTRODUCTION 1 1 The meaning of probability 3 1.1 Classical definition of probability 3 1.2 Statistical definition of probability 9 1.3 Bayesian understanding of probability 10 2 Basic definitions for frequentist statistics and Bayesian inference 15 2.1 Definition of mean, moments and marginal distribution 15 2.2 Worked example: The three-urn problem 24 2.3 Frequentist statistics versus Bayesian inference 28 3 Bayesian inference 33 3.1 Propositions 33 3.2 Selected examples 34 3.3 Ockham s razor 43 4 Combinatorics 47 4.1 Preliminaries 47 4.2 Partitions, binomial and multinomial distributions 52 4.3 Occupation number problems 59 4.4 Geometric and hypergeometric distributions 62 4.5 The negative binomial distribution 66 5 Random walks 71 5.1 First return 71 5.2 First lead 75 5.3 Random walk with absorbing wall 80 6 Limit theorems 83 6.1 Stirling s formula 83 6.2 de Moivre Laplace theorem/local limit theorem 84 6.3 Bernoulli s law of large numbers 86 6.4 Poisson s law 87
Table of vi 7 Continuous distributions 92 7.1 Continuous propositions 92 7.2 Distribution function and probability density functions 93 7.3 Application in statistical physics 96 7.4 Definitions for continuous distributions 98 7.5 Common probability distributions 99 7.6 Order statistic 118 7.7 Transformation of random variables 121 7.8 Characteristic function 124 7.9 Error propagation 131 7.10 Helmert transformation 136 8 The central limit theorem 139 8.1 The theorem 139 8.2 Stable distributions 142 8.3 Proof of the central limit theorem 143 8.4 Markov chain Monte Carlo (MCMC) 144 8.5 The multivariate case 146 9 Poisson processes and waiting times 147 9.1 Stochastic processes 147 9.2 Three ways to generate Poisson points 150 9.3 Waiting time paradox 154 9.4 Order statistic of Poisson processes 156 9.5 Various examples 157 PART II ASSIGNING PROBABILITIES 163 10 Prior probabilities by transformation invariance 165 10.1 Bertrand s paradox revisited 167 10.2 Prior for scale variables 169 10.3 The prior for a location variable 171 10.4 Hyperplane priors 171 10.5 The invariant Riemann measure (Jeffreys prior) 176 11 Testable information and maximum entropy 178 11.1 Discrete case 178 11.2 Properties of the Shannon entropy 182 11.3 Maximum entropy for continuous distributions 194 12 Quantified maximum entropy 201 12.1 The entropic prior 201 12.2 Derivation of the entropic prior 202 12.3 Saddle-point approximation for the normalization 203 12.4 Posterior probability density 204
Table of vii 12.5 Regularization and good data 205 12.6 A technical trick 209 12.7 Application to ill-posed inversion problems 210 13 Global smoothness 215 13.1 A primer on cubic splines 216 13.2 Second derivative prior 219 13.3 First derivative prior 221 13.4 Fisher information prior 221 PART III PARAMETER ESTIMATION 225 14 Bayesian parameter estimation 227 14.1 The estimation problem 227 14.2 Loss and risk function 227 14.3 Confidence intervals 231 14.4 Examples 231 15 Frequentist parameter estimation 236 15.1 Unbiased estimators 236 15.2 The maximum likelihood estimator 237 15.3 Examples 237 15.4 Stopping criteria for experiments 241 15.5 Is unbiasedness desirable at all? 245 15.6 Least-squares fitting 246 16 The Cramer Rao inequality 248 16.1 Lower bound on the variance 248 16.2 Examples 249 16.3 Admissibility of the Cramer Rao limit 251 PART IV TESTING HYPOTHESES 253 17 The Bayesian way 255 17.1 Some illustrative examples 256 17.2 Independent measurements with Gaussian noise 262 18 The frequentist approach 276 18.1 Introduction 276 18.2 Neyman Pearson lemma 281 19 Sampling distributions 284 19.1 Mean and median of i.i.d. random variables 284 19.2 Mean and variance of Gaussian samples 294 19.3 z-statistic 297 19.4 Student s t-statistic 299 19.5 Fisher Snedecor F -statistic 302
Table of viii 19.6 Chi-squared in case of missing parameters 305 19.7 Common hypothesis tests 308 20 Comparison of Bayesian and frequentist hypothesis tests 324 20.1 Prior knowledge is prior data 324 20.2 Dependence on the stopping criterion 325 PART V REAL-WORLD APPLICATIONS 331 21 Regression 333 21.1 Linear regression 334 21.2 Models with nonlinear parameter dependence 350 21.3 Errors in all variables 353 22 Consistent inference on inconsistent data 364 22.1 Erroneously measured uncertainties 364 22.2 Combining incompatible measurements 380 23 Unrecognized signal contributions 396 23.1 The nuclear fission cross-section 239 Pu (n, f ) 396 23.2 Electron temperature in a tokamak edge plasma 399 23.3 Signal background separation 403 24 Change point problems 409 24.1 The Bayesian change point problem 409 24.2 Change points in a binary image 415 24.3 Neural network modelling 420 24.4 Thin film growth detected by Auger analysis 427 25 Function estimation 431 25.1 Deriving trends from observations 432 25.2 Density estimation 439 26 Integral equations 451 26.1 Abel s integral equation 452 26.2 The Laplace transform 456 26.3 The Kramers Kronig relations 459 26.4 Noisy kernels 463 26.5 Deconvolution 465 27 Model selection 470 27.1 Inelastic electron scattering 473 27.2 Signal background separation 474 27.3 Spectral line broadening 478 27.4 Adaptive choice of pivots 481 27.5 Mass spectrometry 484
Table of ix 28 Bayesian experimental design 491 28.1 Overview of the Bayesian approach 491 28.2 Optimality criteria and utility functions 492 28.3 Examples 493 28.4 N-step-ahead designs 504 28.5 Experimental design: Perspective 504 PART VI PROBABILISTIC NUMERICAL TECHNIQUES 507 29 Numerical integration 509 29.1 The deterministic approach 509 29.2 Monte Carlo integration 515 29.3 Beyond the Gaussian approximation 531 30 Monte Carlo methods 537 30.1 Simple sampling 537 30.2 Variance reduction 542 30.3 Markov chain Monte Carlo 544 30.4 Expectation value of the sample mean 555 30.5 Equilibration 560 30.6 Variance of the sample mean 561 30.7 Taming rugged PDFs by tempering 564 30.8 Evidence integral and partition function 568 31 Nested sampling 572 31.1 Motivation 572 31.2 The theory behind nested sampling 579 31.3 Application to the classical ideal gas 584 31.4 Statistical uncertainty 592 31.5 Concluding remarks 594 Appendix A Mathematical compendium 595 Appendix B Selected proofs and derivations 611 Appendix C Symbols and notation 619 References 620 Index 631