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Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® Support [Mīkstie vāki]

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(University of British Columbia, Vancouver)
  • Formāts: Paperback / softback, 488 pages, height x width x depth: 244x170x28 mm, weight: 770 g, Worked examples or Exercises; 132 Line drawings, black and white
  • Izdošanas datums: 20-May-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521150124
  • ISBN-13: 9780521150125
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Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® SupportPalielināt
  • Formāts: Paperback / softback, 488 pages, height x width x depth: 244x170x28 mm, weight: 770 g, Worked examples or Exercises; 132 Line drawings, black and white
  • Izdošanas datums: 20-May-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521150124
  • ISBN-13: 9780521150125
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521150125.html
Keywords:
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® notebooks with solutions to selected problems, additional worked examples, and a Mathematica tutorial are available at www.cambridge.org/9780521150125.

Recenzijas

'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 'The examples are well integrated with the text and are enlightening.' Contemporary Physics 'The book can easily keep the readers amazed and attracted to its content throughout the read and make them want to return back to it recursively. It presents a perfect balance between theoretical inference and a practical know-how approach to Bayesian methods.' Stan Lipovetsky, Technometrics

Papildus informācija

A clear exposition of the underlying concepts, containing large numbers of worked examples and problem sets, first published in 2005.
Preface xiii
Software support xv
Acknowledgements xvii
1 Role of probability theory in science
1(20)
1.1 Scientific inference
1(1)
1.2 Inference requires a probability theory
2(3)
1.2.1 The two rules for manipulating probabilities
4(1)
1.3 Usual form of Bayes' theorem
5(5)
1.3.1 Discrete hypothesis space
5(1)
1.3.2 Continuous hypothesis space
6(1)
1.3.3 Bayes' theorem - model of the learning process
7(1)
1.3.4 Example of the use of Bayes' theorem
8(2)
1.4 Probability and frequency
10(2)
1.4.1 Example: incorporating frequency information
11(1)
1.5 Marginalization
12(3)
1.6 The two basic problems in statistical inference
15(1)
1.7 Advantages of the Bayesian approach
16(1)
1.8 Problems
17(4)
2 Probability theory as extended logic
21(20)
2.1 Overview
21(1)
2.2 Fundamentals of logic
21(4)
2.2.1 Logical propositions
21(1)
2.2.2 Compound propositions
22(1)
2.2.3 Truth tables and Boolean algebra
22(2)
2.2.4 Deductive inference
24(1)
2.2.5 Inductive or plausible inference
25(1)
2.3 Brief history
25(1)
2.4 An adequate set of operations
26(3)
2.4.1 Examination of a logic function
27(2)
2.5 Operations for plausible inference
29(8)
2.5.1 The desiderata of Bayesian probability theory
30(1)
2.5.2 Development of the product rule
30(4)
2.5.3 Development of sum rule
34(2)
2.5.4 Qualitative properties of product and sum rules
36(1)
2.6 Uniqueness of the product and sum rules
37(2)
2.7 Summary
39(1)
2.8 Problems
39(2)
3 The how-to of Bayesian inference
41(31)
3.1 Overview
41(1)
3.2 Basics
41(2)
3.3 Parameter estimation
43(2)
3.4 Nuisance parameters
45(1)
3.5 Model comparison and Occam's razor
45(5)
3.6 Sample spectral line problem
50(2)
3.6.1 Background information
50(2)
3.7 Odds ratio
52(7)
3.7.1 Choice of prior p(T\M1, I)
53(2)
3.7.2 Calculation of p(D\M1, T, I)
55(3)
3.7.3 Calculation of p(D\M2, I)
58(1)
3.7.4 Odds, uniform prior
58(1)
3.7.5 Odds, Jeffreys prior
58(1)
3.8 Parameter estimation problem
59(2)
3.8.1 Sensitivity of odds to Tmax
59(2)
3.9 Lessons
61(2)
3.10 Ignorance priors
63(2)
3.11 Systematic errors
65(4)
3.11.1 Systematic error example
66(3)
3.12 Problems
69(3)
4 Assigning probabilities
72(24)
4.1 Introduction
72(1)
4.2 Binomial distribution
72(7)
4.2.1 Bernoulli's law of large numbers
75(1)
4.2.2 The gambler's coin problem
75(2)
4.2.3 Bayesian analysis of an opinion poll
77(2)
4.3 Multinomial distribution
79(1)
4.4 Can you really answer that question?
80(2)
4.5 Logical versus causal connections
82(1)
4.6 Exchangeable distributions
83(2)
4.7 Poisson distribution
85(4)
4.7.1 Bayesian and frequentist comparison
87(2)
4.8 Constructing likelihood functions
89(4)
4.8.1 Deterministic model
90(1)
4.8.2 Probabilistic model
91(2)
4.9 Summary
93(1)
4.10 Problems
94(2)
5 Frequentist statistical inference
96(43)
5.1 Overview
96(1)
5.2 The concept of a random variable
96(1)
5.3 Sampling theory
97(1)
5.4 Probability distributions
98(2)
5.5 Descriptive properties of distributions
100(5)
5.5.1 Relative line shape measures for distributions
101(1)
5.5.2 Standard random variable
102(1)
5.5.3 Other measures of central tendency and dispersion
103(1)
5.5.4 Median baseline subtraction
104(1)
5.6 Moment generating functions
105(2)
5.7 Some discrete probability distributions
107(6)
5.7.1 Binomial distribution
107(2)
5.7.2 The Poisson distribution
109(3)
5.7.3 Negative binomial distribution
112(1)
5.8 Continuous probability distributions
113(6)
5.8.1 Normal distribution
113(3)
5.8.2 Uniform distribution
116(1)
5.8.3 Gamma distribution
116(1)
5.8.4 Beta distribution
117(1)
5.8.5 Negative exponential distribution
118(1)
5.9 Central Limit Theorem
119(1)
5.10 Bayesian demonstration of the Central Limit Theorem
120(4)
5.11 Distribution of the sample mean
124(1)
5.11.1 Signal averaging example
125(1)
5.12 Transformation of a random Variable
125(2)
5.13 Random and pseudo-random numbers
127(9)
5.13.1 Pseudo-random number generators
131(1)
5.13.2 Tests for randomness
132(4)
5.14 Summary
136(1)
5.15 Problems
137(2)
6 What is a statistic?
139(23)
6.1 Introduction
139(2)
6.2 The x2 distribution
141(2)
6.3 Sample variance S2
143(4)
6.4 The Student's t distribution
147(3)
6.5 F distribution (F-test)
150(2)
6.6 Confidence intervals
152(8)
6.6.1 Variance σ2 known
152(4)
6.6.2 Confidence intervals for μ, unknown variance
156(2)
6.6.3 Confidence intervals: difference of two means
158(1)
6.6.4 Confidence intervals for σ2
159(1)
6.6.5 Confidence intervals: ratio of two variances
159(1)
6.7 Summary
160(1)
6.8 Problems
161(1)
7 Frequentist hypothesis testing
162(22)
7.1 Overview
162(1)
7.2 Basic idea
162(10)
7.2.1 Hypothesis testing with the x2 statistic
163(4)
7.2.2 Hypothesis test on the difference of two means
167(3)
7.2.3 One-sided and two-sided hypothesis tests
170(2)
7.3 Are two distributions the same?
172(5)
7.3.1 Pearson x2 goodness-of-fit test
173(4)
7.3.2 Comparison of two-binned data sets
177(1)
7.4 Problem with frequentist hypothesis testing
177(4)
7.4.1 Bayesian resolution to optional stopping problem
179(2)
7.5 Problems
181(3)
8 Maximum entropy probabilities
184(28)
8.1 Overview
184(1)
8.2 The maximum entropy principle
185(1)
8.3 Shannon's theorem
186(1)
8.4 Alternative justification of MaxEnt
187(3)
8.5 Generalizing MaxEnt
190(1)
8.5.1 Incorporating a prior
190(1)
8.5.2 Continuous probability distributions
191(1)
8.6 How to apply the MaxEnt principle
191(1)
8.6.1 Lagrange multipliers of variational calculus
191(1)
8.7 MaxEnt distributions
192(11)
8.7.1 General properties
192(2)
8.7.2 Uniform distribution
194(1)
8.7.3 Exponential distribution
195(2)
8.7.4 Normal and truncated Gaussian distributions
197(5)
8.7.5 Multivariate Gaussian distribution
202(1)
8.8 MaxEnt image reconstruction
203(5)
8.8.1 The kangaroo justification
203(3)
8.8.2 MaxEnt for uncertain constraints
206(2)
8.9 Pixon multiresolution image reconstruction
208(3)
8.10 Problems
211(1)
9 Bayesian inference with Gaussian errors
212(31)
9.1 Overview
212(1)
9.2 Bayesian estimate of a mean
212(15)
9.2.1 Mean: known noise σ
213(4)
9.2.2 Mean: known noise, unequal σ
217(1)
9.2.3 Mean: unknown noise σ
218(6)
9.2.4 Bayesian estimate of σ
224(3)
9.3 Is the signal variable?
227(1)
9.4 Comparison of two independent samples
228(12)
9.4.1 Do the samples differ?
230(3)
9.4.2 How do the samples differ?
233(1)
9.4.3 Results
233(3)
9.4.4 The difference in means
236(1)
9.4.5 Ratio of the standard deviations
237(2)
9.4.6 Effect of the prior ranges
239(1)
9.5 Summary
240(1)
9.6 Problems
241(2)
10 Linear model fitting (Gaussian errors)
243(44)
10.1 Overview
243(1)
10.2 Parameter estimation
244(12)
10.2.1 Most probable amplitudes
249(4)
10.2.2 More powerful matrix formulation
253(3)
10.3 Regression analysis
256(1)
10.4 The posterior is a Gaussian
257(7)
10.4.1 Joint credible regions
260(4)
10.5 Model parameter errors
264(9)
10.5.1 Marginalization and the covariance matrix
264(4)
10.5.2 Correlation coefficient
268(4)
10.5.3 More on model parameter errors
272(1)
10.6 Correlated data errors
273(2)
10.7 Model comparison with Gaussian posteriors
275(4)
10.8 Frequentist testing and errors
279(4)
10.8.1 Other model comparison methods
281(2)
10.9 Summary
283(1)
10.10 Problems
284(3)
11 Nonlinear model fitting
287(25)
11.1 Introduction
287(1)
11.2 Asymptotic normal approximation
288(3)
11.3 Laplacian approximations
291(3)
11.3.1 Bayes factor
291(2)
11.3.2 Marginal parameter posteriors
293(1)
11.4 Finding the most probable parameters
294(4)
11.4.1 Simulated annealing
296(1)
11.4.2 Genetic algorithm
297(1)
11.5 Iterative linearization
298(4)
11.5.1 Levenberg-Marquardt method
300(1)
11.5.2 Marquardt's recipe
301(1)
11.6 Mathematica example
302(5)
11.6.1 Model comparison
304(2)
11.6.2 Marginal and projected distributions
306(1)
11.7 Errors in both coordinates
307(2)
11.8 Summary
309(1)
11.9 Problems
309(3)
12 Markov chain Monte Carlo
312(40)
12.1 Overview
312(1)
12.2 Metropolis-Hastings algorithm
313(6)
12.3 Why does Metropolis-Hastings work?
319(2)
12.4 Simulated tempering
321(1)
12.5 Parallel tempering
321(1)
12.6 Example
322(4)
12.7 Model comparison
326(4)
12.8 Towards an automated MCMC
330(1)
12.9 Extrasolar planet example
331(11)
12.9.1 Model probabilities
335(2)
12.9.2 Results
337(5)
12.10 MCMC robust summary statistic
342(4)
12.11 Summary
346(3)
12.12 Problems
349(3)
13 Bayesian revolution in spectral analysis
352(24)
13.1 Overview
352(1)
13.2 New insights on the periodogram
352(6)
13.2.1 How to compute p(f|D, I)
356(2)
13.3 Strong prior signal model
358(2)
13.4 No specific prior signal model
360(5)
13.4.1 X-ray astronomy example
362(1)
13.4.2 Radio astronomy example
363(2)
13.5 Generalized Lomb-Scargle periodogram
365(5)
13.5.1 Relationship to Lomb-Scargle periodogram
367(1)
13.5.2 Example
367(3)
13.6 Non-uniform sampling
370(3)
13.7 Problems
373(3)
14 Bayesian inference with Poisson sampling
376(13)
14.1 Overview
376(1)
14.2 Infer a Poisson rate
377(2)
14.2.1 Summary of posterior
378(1)
14.3 Signal + known background
379(1)
14.4 Analysis of ON/OFF measurements
380(6)
14.4.1 Estimating the source rate
381(3)
14.4.2 Source detection question
384(2)
14.5 Time-varying Poisson rate
386(2)
14.6 Problems
388(1)
Appendix A Singular value decomposition
389(3)
Appendix B Discrete Fourier Transforms
392(42)
B.1 Overview
392(1)
B.2 Orthogonal and orthonormal functions
392(2)
B.3 Fourier series and integral transform
394(4)
B.3.1 Fourier series
395(1)
B.3.2 Fourier transform
396(2)
B.4 Convolution and correlation
398(5)
B.4.1 Convolution theorem
399(1)
B.4.2 Correlation theorem
400(1)
B.4.3 Importance of convolution in science
401(2)
B.5 Waveform sampling
403(1)
B.6 Nyquist sampling theorem
404(3)
B.6.1 Astronomy example
406(1)
B.7 Discrete Fourier Transform
407(4)
B.7.1 Graphical development
407(2)
B.7.2 Mathematical development of the DFT
409(1)
B.7.3 Inverse DFT
410(1)
B.8 Applying the DFT
411(4)
B.8.1 DFT as an approximate Fourier transform
411(2)
B.8.2 Inverse discrete Fourier transform
413(2)
B.9 The Fast Fourier Transform
415(2)
B.10 Discrete convolution and correlation
417(5)
B.10.1 Deconvolving a noisy signal
418(2)
B.10.2 Deconvolution with an optimal Weiner filter
420(1)
B.10.3 Treatment of end effects by zero padding
421(1)
B.11 Accurate amplitudes by zero padding
422(2)
B.12 Power-spectrum estimation
424(4)
B.12.1 Parseval's theorem and power spectral density
424(1)
B.12.2 Periodogram power-spectrum estimation
425(1)
B.12.3 Correlation spectrum estimation
426(2)
B.13 Discrete power spectral density estimation
428(4)
B.13.1 Discrete form of Parseval's theorem
428(1)
B.13.2 One-sided discrete power spectral density
429(1)
B.13.3 Variance of periodogram estimate
429(2)
B.13.4 Yule's stochastic spectrum estimation model
431(1)
B.13.5 Reduction of periodogram variance
431(1)
B.14 Problems
432(2)
Appendix C Difference in two samples
434(11)
C.1 Outline
434(1)
C.2 Probabilities of the four hypotheses
434(5)
C.2.1 Evaluation of p(C, S|D1, D2, I)
434(2)
C.2.2 Evaluation of p(C, S|D1, D2, I)
436(2)
C.2.3 Evaluation of p(C, S|D1, D2, I)
438(1)
C.2.4 Evaluation of p(C, S|D1, D2, I)
439(1)
C.3 The difference in the means
439(3)
C.3.1 The two-sample problem
440(1)
C.3.2 The Behrens-Fisher problem
441(1)
C.4 The ratio of the standard deviations
442(3)
C.4.1 Estimating the ratio, given the means are the same
442(1)
C.4.2 Estimating the ratio, given the means are different
443(2)
Appendix D Poisson ON/OFF details
445(5)
D.1 Derivation of p(s|Non, I)
445(3)
D. 1.1 Evaluation of Num
446(1)
D.1.2 Evaluation of Den
447(1)
D.2 Derivation of the Bayes factor B{s+b, b}
448(2)
Appendix E Multivariate Gaussian from maximum entropy
450(5)
References 455(6)
Index 461
Phil Gregory is Professor Emeritus at the Department of Physics and Astronomy at the University of British Columbia.