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Bayesian Logical Data Analysis for the Physical Sciences : A Comparative Approach with Mathematica (R) Support - Phil Gregory

Bayesian Logical Data Analysis for the Physical Sciences

A Comparative Approach with Mathematica (R) Support


Published: 1st June 2010
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Increasingly, researchers in many branches of science are coming into contact with Bayesian statistics or Bayesian probability theory. By encompassing both inductive and deductive logic, Bayesian analysis can improve model parameter estimates by many orders of magnitude. It provides a simple and unified approach to all data analysis problems, allowing the experimenter to assign probabilities to competing hypotheses of interest, on the basis of the current state of knowledge. This book provides a clear exposition of the underlying concepts with large numbers of worked examples and problem sets. The book also discusses numerical techniques for implementing the Bayesian calculations, including an introduction to Markov Chain Monte-Carlo integration and linear and nonlinear least-squares analysis seen from a Bayesian perspective. In addition, background material is provided in appendices and supporting Mathematica notebooks are available, providing an easy learning route for upper-undergraduates, graduate students, or any serious researcher in physical sciences or engineering.

'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

Prefacep. xiii
Software supportp. xv
Acknowledgementsp. xvii
Role of probability theory in sciencep. 1
Scientific inferencep. 1
Inference requires a probability theoryp. 2
The two rules for manipulating probabilitiesp. 4
Usual form of Bayes' theoremp. 5
Discrete hypothesis spacep. 5
Continuous hypothesis spacep. 6
Bayes' theorem - model of the learning processp. 7
Example of the use of Bayes' theoremp. 8
Probability and frequencyp. 10
Example: incorporating frequency informationp. 11
Marginalizationp. 12
The two basic problems in statistical inferencep. 15
Advantages of the Bayesian approachp. 16
Problemsp. 17
Probability theory as extended logicp. 21
Overviewp. 21
Fundamentals of logicp. 21
Logical propositionsp. 21
Compound propositionsp. 22
Truth tables and Boolean algebrap. 22
Deductive inferencep. 24
Inductive or plausible inferencep. 25
Brief historyp. 25
An adequate set of operationsp. 26
Examination of a logic functionp. 27
Operations for plausible inferencep. 29
The desiderata of Bayesian probability theoryp. 30
Development of the product rulep. 30
Development of sum rulep. 34
Qualitative properties of product and sum rulesp. 36
Uniqueness of the product and sum rulesp. 37
Summaryp. 39
Problemsp. 39
The how-to of Bayesian inferencep. 41
Overviewp. 41
Basicsp. 41
Parameter estimationp. 43
Nuisance parametersp. 45
Model comparison and Occam's razorp. 45
Sample spectral line problemp. 50
Background informationp. 50
Odds ratiop. 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 priorp. 58
Odds, Jeffreys priorp. 58
Parameter estimation problemp. 59
Sensitivity of odds to T[subscript max]p. 59
Lessonsp. 61
Ignorance priorsp. 63
Systematic errorsp. 65
Systematic error examplep. 66
Problemsp. 69
Assigning probabilitiesp. 72
Introductionp. 72
Binomial distributionp. 72
Bernoulli's law of large numbersp. 75
The gambler's coin problemp. 75
Bayesian analysis of an opinion pollp. 77
Multinomial distributionp. 79
Can you really answer that question?p. 80
Logical versus causal connectionsp. 82
Exchangeable distributionsp. 83
Poisson distributionp. 85
Bayesian and frequentist comparisonp. 87
Constructing likelihood functionsp. 89
Deterministic modelp. 90
Probabilistic modelp. 91
Summaryp. 93
Problemsp. 94
Frequentist statistical inferencep. 96
Overviewp. 96
The concept of a random variablep. 96
Sampling theoryp. 97
Probability distributionsp. 98
Descriptive properties of distributionsp. 100
Relative line shape measures for distributionsp. 101
Standard random variablep. 102
Other measures of central tendency and dispersionp. 103
Median baseline subtractionp. 104
Moment generating functionsp. 105
Some discrete probability distributionsp. 107
Binomial distributionp. 107
The Poisson distributionp. 109
Negative binomial distributionp. 112
Continuous probability distributionsp. 113
Normal distributionp. 113
Uniform distributionp. 116
Gamma distributionp. 116
Beta distributionp. 117
Negative exponential distributionp. 118
Central Limit Theoremp. 119
Bayesian demonstration of the Central Limit Theoremp. 120
Distribution of the sample meanp. 124
Signal averaging examplep. 125
Transformation of a random variablep. 125
Random and pseudo-random numberp. 127
Pseudo-random number generatorsp. 131
Tests for randomnessp. 132
Summaryp. 136
Problemsp. 137
What is a statistic?p. 139
Introductionp. 139
The x[superscript 2] distributionp. 141
Sample variance S[superscript 2]p. 143
The Student's t distributionp. 147
F distribution (F-test)p. 150
Confidence intervalsp. 152
Variance [sigma superscript 2] knownp. 152
Confidence intervals for [mu], unknown variancep. 156
Confidence intervals: difference of two meansp. 158
Confidence intervals for [sigma superscript 2]p. 159
Confidence intervals: ratio of two variancesp. 159
Summaryp. 160
Problemsp. 161
Frequentist hypothesis testingp. 162
Overviewp. 162
Basic ideap. 162
Hypothesis testing with the x[superscript 2] statisticp. 163
Hypothesis test on the difference of two meansp. 167
One-sided and two-sided hypothesis testsp. 170
Are two distributions the same?p. 172
Pearson x[superscript 2] goodness-of-fit testp. 173
Comparison of two-binned data setsp. 177
Problem with frequentist hypothesis testingp. 177
Bayesian resolution to optional stopping problemp. 179
Problemsp. 181
Maximum entropy probabilitiesp. 184
Overviewp. 184
The maximum entropy principlep. 185
Shannon's theoremp. 186
Alternative justification of MaxEntp. 187
Generalizing MaxEntp. 190
Incorporating a priorp. 190
Continuous probability distributionsp. 191
How to apply the MaxEnt principlep. 191
Lagrange multipliers of variational calculusp. 191
MaxEnt distributionsp. 192
General propertiesp. 192
Uniform distributionp. 194
Exponential distributionp. 195
Normal and truncated Gaussian distributionsp. 197
Multivariate Gaussian distributionp. 202
MaxEnt image reconstructionp. 203
The kangaroo justificationp. 203
MaxEnt for uncertain constraintsp. 206
Pixon multiresolution image reconstructionp. 208
Problemsp. 211
Bayesian inference with Gaussian errorsp. 212
Overviewp. 212
Bayesian estimate of a meanp. 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 samplesp. 228
Do the samples differ?p. 230
How do the samples differ?p. 233
Resultsp. 233
The difference in meansp. 236
Ratio of the standard deviationsp. 237
Effect of the prior rangesp. 239
Summaryp. 240
Problemsp. 241
Linear model fitting (Gaussian errors)p. 243
Overviewp. 243
Parameter estimationp. 244
Most probable amplitudesp. 249
More powerful matrix formulationp. 253
Regression analysisp. 256
The posterior is a Gaussianp. 257
Joint credible regionsp. 260
Model parameter errorsp. 264
Marginalization and the covariance matrixp. 264
Correlation coefficientp. 268
More on model parameter errorsp. 272
Correlated data errorsp. 273
Model comparison with Gaussian posteriorsp. 275
Frequentist testing and errorsp. 279
Other model comparison methodsp. 281
Summaryp. 283
Problemsp. 284
Nonlinear model fittingp. 287
Introductionp. 287
Asymptotic normal approximationp. 288
Laplacian approximationsp. 291
Bayes factorp. 291
Marginal parameter posteriorsp. 293
Finding the most probable parametersp. 294
Simulated annealingp. 296
Genetic algorithmp. 297
Iterative linearizationp. 298
Levenberg-Marquardt methodp. 300
Marquardt's recipep. 301
Mathematica examplep. 302
Model comparisonp. 304
Marginal and projected distributionsp. 306
Errors in both coordinatesp. 307
Summaryp. 309
Problemsp. 309
Markov chain Monte Carlop. 312
Overviewp. 312
Metropolis-Hastings algorithmp. 313
Why does Metropolis-Hastings work?p. 319
Simulated temperingp. 321
Parallel temperingp. 321
Examplep. 322
Model comparisonp. 326
Towards an automated MCMCp. 330
Extrasolar planet examplep. 331
Model probabilitiesp. 335
Resultsp. 337
MCMC robust summary statisticp. 342
Summaryp. 346
Problemsp. 349
Bayesian revolution in spectral analysisp. 352
Overviewp. 352
New insights on the periodogramp. 352
How to compute p(f[vertical bar]D,I)p. 356
Strong prior signal modelp. 358
No specific prior signal modelp. 360
X-ray astronomy examplep. 362
Radio astronomy examplep. 363
Generalized Lomb-Scargle periodogramp. 365
Relationship to Lomb-Scargle periodogramp. 367
Examplep. 367
Non-uniform samplingp. 370
Problemsp. 373
Bayesian inference with Poisson samplingp. 376
Overviewp. 376
Infer a Poisson ratep. 377
Summary of posteriorp. 378
Signal + known backgroundp. 379
Analysis of ON/OFF measurementsp. 380
Estimating the source ratep. 381
Source detection questionp. 384
Time-varying Poisson ratep. 386
Problemsp. 388
Singular value decompositionp. 389
Discrete Fourier Transformsp. 392
Overviewp. 392
Orthogonal and orthonormal functionsp. 392
Fourier series and integral transformp. 394
Fourier seriesp. 395
Fourier transformp. 396
Convolution and correlationp. 398
Convolution theoremp. 399
Correlation theoremp. 400
Importance of convolution in sciencep. 401
Waveform samplingp. 403
Nyquist sampling theoremp. 404
Astronomy examplep. 406
Discrete Fourier Transformp. 407
Graphical developmentp. 407
Mathematical development of the DFTp. 409
Inverse DFTp. 410
Applying the DFTp. 411
DFT as an approximate Fourier transformp. 411
Inverse discrete Fourier transformp. 413
The Fast Fourier Transformp. 415
Discrete convolution and correlationp. 417
Deconvolving a noisy signalp. 418
Deconvolution with an optimal Weiner filterp. 420
Treatment of end effects by zero paddingp. 421
Accurate amplitudes by zero paddingp. 422
Power-spectrum estimationp. 424
Parseval's theorem and power spectral densityp. 424
Periodogram power-spectrum estimationp. 425
Correlation spectrum estimationp. 426
Discrete power spectral density estimationp. 428
Discrete form of Parseval's theoremp. 428
One-sided discrete power spectral densityp. 429
Variance of periodogram estimatep. 429
Yule's stochastic spectrum estimation modelp. 431
Reduction of periodogram variancep. 431
Problemsp. 432
Difference in two samplesp. 434
Outlinep. 434
Probabilities of the four hypothesesp. 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 meansp. 439
The two-sample problemp. 440
The Behrens-Fisher problemp. 441
The ratio of the standard deviationsp. 442
Estimating the ratio, given the means are the samep. 442
Estimating the ratio, given the means are differentp. 443
Poisson ON/OFF detailsp. 445
Derivation of p(s[vertical bar]N[subscript on], I)p. 445
Evaluation of Nump. 446
Evaluation of Denp. 447
Derivation of the Bayes factor B[subscript {s+b,b}]p. 448
Multivariate Gaussian from maximum entropyp. 450
Referencesp. 455
Indexp. 461
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521841504
ISBN-10: 052184150X
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 488
Published: 1st June 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 24.7 x 17.4  x 2.7
Weight (kg): 1.06