Bayesian Logical Data Analysis for the Physical Sciences

A Comparative Approach with Mathematica Support

By: Phil Gregory

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Hardcover | 27 April 2005

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Published: 20th May 2010

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Bayesian inference provides a simple and unified approach to data analysis, allowing experimenters to assign probabilities to competing hypotheses of interest, on the basis of the current state of knowledge. By incorporating relevant prior information, it can sometimes improve model parameter estimates by many orders of magnitude. This book provides a clear exposition of the underlying concepts with many worked examples and problem sets. It also discusses implementation, including an introduction to Markov chain Monte-Carlo integration and linear and nonlinear model fitting. Particularly extensive coverage of spectral analysis (detecting and measuring periodic signals) includes a self-contained introduction to Fourier and discrete Fourier methods. There is a chapter devoted to Bayesian inference with Poisson sampling, and three chapters on frequentist methods help to bridge the gap between the frequentist and Bayesian approaches. Supporting Mathematica(R) notebooks with solutions to selected problems, additional worked examples, and a Mathematica tutorial are available at www.cambridge.org/9780521150125.

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'As well as the usual topics to be found in a text on Bayesian inference, chapters are included on frequentist inference (for contrast), non-linear model fitting, spectral analysis and Poisson sampling.' Zentralblatt MATH ' ... a clearly written guide to applied statistical analysis by using SPSS software. it is filled with examples and exercises that help readers to understand the types of problem that the techniques can address. If you are an SPSS enthusiast, a beginner or not, you will find this new edition a satisfying source of valuable infromation.' Journal of the RSS

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Non-Fiction Mathematics Probability & Statistics Geometry Science Astronomy, Space & Time Theoretical & Mathematical Astronomy

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Preface	p. xiii
Software support	p. xv
Acknowledgements	p. xvii
Role of probability theory in science	p. 1
Scientific inference	p. 1
Inference requires a probability theory	p. 2
The two rules for manipulating probabilities	p. 4
Usual form of Bayes' theorem	p. 5
Discrete hypothesis space	p. 5
Continuous hypothesis space	p. 6
Bayes' theorem - model of the learning process	p. 7
Example of the use of Bayes' theorem	p. 8
Probability and frequency	p. 10
Example: incorporating frequency information	p. 11
Marginalization	p. 12
The two basic problems in statistical inference	p. 15
Advantages of the Bayesian approach	p. 16
Problems	p. 17
Probability theory as extended logic	p. 21
Overview	p. 21
Fundamentals of logic	p. 21
Logical propositions	p. 21
Compound propositions	p. 22
Truth tables and Boolean algebra	p. 22
Deductive inference	p. 24
Inductive or plausible inference	p. 25
Brief history	p. 25
An adequate set of operations	p. 26
Examination of a logic function	p. 27
Operations for plausible inference	p. 29
The desiderata of Bayesian probability theory	p. 30
Development of the product rule	p. 30
Development of sum rule	p. 34
Qualitative properties of product and sum rules	p. 36
Uniqueness of the product and sum rules	p. 37
Summary	p. 39
Problems	p. 39
The how-to of Bayesian inference	p. 41
Overview	p. 41
Basics	p. 41
Parameter estimation	p. 43
Nuisance parameters	p. 45
Model comparison and Occam's razor	p. 45
Sample spectral line problem	p. 50
Background information	p. 50
Odds ratio	p. 52
Choice of prior p(T[vertical bar]M[subscript 1], I)	p. 53
Calculation of p(D[vertical bar]M[subscript 1], T, I)	p. 55
Calculation of p(D[vertical bar]M[subscript 2], I)	p. 58
Odds, uniform prior	p. 58
Odds, Jeffreys prior	p. 58
Parameter estimation problem	p. 59
Sensitivity of odds to T[subscript max]	p. 59
Lessons	p. 61
Ignorance priors	p. 63
Systematic errors	p. 65
Systematic error example	p. 66
Problems	p. 69
Assigning probabilities	p. 72
Introduction	p. 72
Binomial distribution	p. 72
Bernoulli's law of large numbers	p. 75
The gambler's coin problem	p. 75
Bayesian analysis of an opinion poll	p. 77
Multinomial distribution	p. 79
Can you really answer that question?	p. 80
Logical versus causal connections	p. 82
Exchangeable distributions	p. 83
Poisson distribution	p. 85
Bayesian and frequentist comparison	p. 87
Constructing likelihood functions	p. 89
Deterministic model	p. 90
Probabilistic model	p. 91
Summary	p. 93
Problems	p. 94
Frequentist statistical inference	p. 96
Overview	p. 96
The concept of a random variable	p. 96
Sampling theory	p. 97
Probability distributions	p. 98
Descriptive properties of distributions	p. 100
Relative line shape measures for distributions	p. 101
Standard random variable	p. 102
Other measures of central tendency and dispersion	p. 103
Median baseline subtraction	p. 104
Moment generating functions	p. 105
Some discrete probability distributions	p. 107
Binomial distribution	p. 107
The Poisson distribution	p. 109
Negative binomial distribution	p. 112
Continuous probability distributions	p. 113
Normal distribution	p. 113
Uniform distribution	p. 116
Gamma distribution	p. 116
Beta distribution	p. 117
Negative exponential distribution	p. 118
Central Limit Theorem	p. 119
Bayesian demonstration of the Central Limit Theorem	p. 120
Distribution of the sample mean	p. 124
Signal averaging example	p. 125
Transformation of a random variable	p. 125
Random and pseudo-random number	p. 127
Pseudo-random number generators	p. 131
Tests for randomness	p. 132
Summary	p. 136
Problems	p. 137
What is a statistic?	p. 139
Introduction	p. 139
The x[superscript 2] distribution	p. 141
Sample variance S[superscript 2]	p. 143
The Student's t distribution	p. 147
F distribution (F-test)	p. 150
Confidence intervals	p. 152
Variance [sigma superscript 2] known	p. 152
Confidence intervals for [mu], unknown variance	p. 156
Confidence intervals: difference of two means	p. 158
Confidence intervals for [sigma superscript 2]	p. 159
Confidence intervals: ratio of two variances	p. 159
Summary	p. 160
Problems	p. 161
Frequentist hypothesis testing	p. 162
Overview	p. 162
Basic idea	p. 162
Hypothesis testing with the x[superscript 2] statistic	p. 163
Hypothesis test on the difference of two means	p. 167
One-sided and two-sided hypothesis tests	p. 170
Are two distributions the same?	p. 172
Pearson x[superscript 2] goodness-of-fit test	p. 173
Comparison of two-binned data sets	p. 177
Problem with frequentist hypothesis testing	p. 177
Bayesian resolution to optional stopping problem	p. 179
Problems	p. 181
Maximum entropy probabilities	p. 184
Overview	p. 184
The maximum entropy principle	p. 185
Shannon's theorem	p. 186
Alternative justification of MaxEnt	p. 187
Generalizing MaxEnt	p. 190
Incorporating a prior	p. 190
Continuous probability distributions	p. 191
How to apply the MaxEnt principle	p. 191
Lagrange multipliers of variational calculus	p. 191
MaxEnt distributions	p. 192
General properties	p. 192
Uniform distribution	p. 194
Exponential distribution	p. 195
Normal and truncated Gaussian distributions	p. 197
Multivariate Gaussian distribution	p. 202
MaxEnt image reconstruction	p. 203
The kangaroo justification	p. 203
MaxEnt for uncertain constraints	p. 206
Pixon multiresolution image reconstruction	p. 208
Problems	p. 211
Bayesian inference with Gaussian errors	p. 212
Overview	p. 212
Bayesian estimate of a mean	p. 212
Mean: known noise [sigma]	p. 213
Mean: known noise, unequal [sigma]	p. 217
Mean: unknown noise [sigma]	p. 218
Bayesian estimate of [sigma]	p. 224
Is the signal variable?	p. 227
Comparison of two independent samples	p. 228
Do the samples differ?	p. 230
How do the samples differ?	p. 233
Results	p. 233
The difference in means	p. 236
Ratio of the standard deviations	p. 237
Effect of the prior ranges	p. 239
Summary	p. 240
Problems	p. 241
Linear model fitting (Gaussian errors)	p. 243
Overview	p. 243
Parameter estimation	p. 244
Most probable amplitudes	p. 249
More powerful matrix formulation	p. 253
Regression analysis	p. 256
The posterior is a Gaussian	p. 257
Joint credible regions	p. 260
Model parameter errors	p. 264
Marginalization and the covariance matrix	p. 264
Correlation coefficient	p. 268
More on model parameter errors	p. 272
Correlated data errors	p. 273
Model comparison with Gaussian posteriors	p. 275
Frequentist testing and errors	p. 279
Other model comparison methods	p. 281
Summary	p. 283
Problems	p. 284
Nonlinear model fitting	p. 287
Introduction	p. 287
Asymptotic normal approximation	p. 288
Laplacian approximations	p. 291
Bayes factor	p. 291
Marginal parameter posteriors	p. 293
Finding the most probable parameters	p. 294
Simulated annealing	p. 296
Genetic algorithm	p. 297
Iterative linearization	p. 298
Levenberg-Marquardt method	p. 300
Marquardt's recipe	p. 301
Mathematica example	p. 302
Model comparison	p. 304
Marginal and projected distributions	p. 306
Errors in both coordinates	p. 307
Summary	p. 309
Problems	p. 309
Markov chain Monte Carlo	p. 312
Overview	p. 312
Metropolis-Hastings algorithm	p. 313
Why does Metropolis-Hastings work?	p. 319
Simulated tempering	p. 321
Parallel tempering	p. 321
Example	p. 322
Model comparison	p. 326
Towards an automated MCMC	p. 330
Extrasolar planet example	p. 331
Model probabilities	p. 335
Results	p. 337
MCMC robust summary statistic	p. 342
Summary	p. 346
Problems	p. 349
Bayesian revolution in spectral analysis	p. 352
Overview	p. 352
New insights on the periodogram	p. 352
How to compute p(f[vertical bar]D,I)	p. 356
Strong prior signal model	p. 358
No specific prior signal model	p. 360
X-ray astronomy example	p. 362
Radio astronomy example	p. 363
Generalized Lomb-Scargle periodogram	p. 365
Relationship to Lomb-Scargle periodogram	p. 367
Example	p. 367
Non-uniform sampling	p. 370
Problems	p. 373
Bayesian inference with Poisson sampling	p. 376
Overview	p. 376
Infer a Poisson rate	p. 377
Summary of posterior	p. 378
Signal + known background	p. 379
Analysis of ON/OFF measurements	p. 380
Estimating the source rate	p. 381
Source detection question	p. 384
Time-varying Poisson rate	p. 386
Problems	p. 388
Singular value decomposition	p. 389
Discrete Fourier Transforms	p. 392
Overview	p. 392
Orthogonal and orthonormal functions	p. 392
Fourier series and integral transform	p. 394
Fourier series	p. 395
Fourier transform	p. 396
Convolution and correlation	p. 398
Convolution theorem	p. 399
Correlation theorem	p. 400
Importance of convolution in science	p. 401
Waveform sampling	p. 403
Nyquist sampling theorem	p. 404
Astronomy example	p. 406
Discrete Fourier Transform	p. 407
Graphical development	p. 407
Mathematical development of the DFT	p. 409
Inverse DFT	p. 410
Applying the DFT	p. 411
DFT as an approximate Fourier transform	p. 411
Inverse discrete Fourier transform	p. 413
The Fast Fourier Transform	p. 415
Discrete convolution and correlation	p. 417
Deconvolving a noisy signal	p. 418
Deconvolution with an optimal Weiner filter	p. 420
Treatment of end effects by zero padding	p. 421
Accurate amplitudes by zero padding	p. 422
Power-spectrum estimation	p. 424
Parseval's theorem and power spectral density	p. 424
Periodogram power-spectrum estimation	p. 425
Correlation spectrum estimation	p. 426
Discrete power spectral density estimation	p. 428
Discrete form of Parseval's theorem	p. 428
One-sided discrete power spectral density	p. 429
Variance of periodogram estimate	p. 429
Yule's stochastic spectrum estimation model	p. 431
Reduction of periodogram variance	p. 431
Problems	p. 432
Difference in two samples	p. 434
Outline	p. 434
Probabilities of the four hypotheses	p. 434
Evaluation of p(C, S[vertical bar]D[subscript 1], D[subscript 2], I)	p. 434
Evaluation of p(C, S[vertical bar]D[subscript 1], D[subscript 2], I)	p. 436
Evaluation of p(C, S[vertical bar]D[subscript 1], D[subscript 2], I)	p. 438
Evaluation of p(C, S[vertical bar]D[subscript 1], D[subscript 2], I)	p. 439
The difference in the means	p. 439
The two-sample problem	p. 440
The Behrens-Fisher problem	p. 441
The ratio of the standard deviations	p. 442
Estimating the ratio, given the means are the same	p. 442
Estimating the ratio, given the means are different	p. 443
Poisson ON/OFF details	p. 445
Derivation of p(s[vertical bar]N[subscript on], I)	p. 445
Evaluation of Num	p. 446
Evaluation of Den	p. 447
Derivation of the Bayes factor B[subscript {s+b,b}]	p. 448
Multivariate Gaussian from maximum entropy	p. 450
References	p. 455
Index	p. 461
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Bayesian Logical Data Analysis for the Physical Sciences

A Comparative Approach with Mathematica Support

At a Glance

New Edition

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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