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E-raamat: Bayesian Inference: Data Evaluation and Decisions

  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Oct-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319416441
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Bayesian Inference: Data Evaluation and DecisionsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 18-Oct-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319416441
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This new edition offers a comprehensive introduction to the analysis of data using Bayes rule. It generalizes Gaussian error intervals to situations in which the data follow distributions other than Gaussian. This is particularly useful when the observed parameter is barely above the background or the histogram of multiparametric data contains many empty bins, so that the determination of the validity of a theory cannot be based on the chi-squared-criterion. In addition to the solutions of practical problems, this approach provides an epistemic insight: the logic of quantum mechanics is obtained as the logic of unbiased inference from counting data. New sections feature factorizing parameters, commuting parameters, observables in quantum mechanics, the art of fitting with coherent and with incoherent alternatives and fitting with multinomial distribution. Additional problems and examples help deepen the knowledge. Requiring no knowledge of quantum mechanics, the book is written on

introductory level, with many examples and exercises, for advanced undergraduate and graduate students in the physical sciences, planning to, or working in, fields such as medical physics, nuclear physics, quantum mechanics, and chaos.

Knowledge an Logic.- Bayes" Theorem.- Probable and Improbable Data.- Descriptions of Distributions I: Real x.- Description of Distributions II: Natural x .- Form Invariance I.- Examples of Invariant Measures.- A Linear Representation of Form Invariance.- Going Beyond Form Invariance: The Geometric Prior.- Inferring the Mean or Standard Deviation.- Form Invariance II: Natural x .- Item Response Theory.- On the Art of Fitting .- Problems and Solutions.- Description of Distributions I.- Real x.- Form Invariance I.- Beyond Form Invariance: The Geometric Prior.- Inferring Mean or Standard Deviation.- Form Invariance II: Natural x .- Item Response Theory.- On the Art of Fitting.
1 Knowledge and Logic
1(10)
1.1 Knowledge
2(2)
1.2 Logic
4(1)
1.3 Ignorance
5(2)
1.4 Decisions
7(4)
References
7(4)
2 Bayes' Theorem
11(16)
2.1 Derivation of the Theorem
11(2)
2.2 Transformations
13(2)
2.3 The Concept of Form Invariance
15(1)
2.4 Many Events
16(5)
2.5 Improper Distributions
21(2)
2.6 Shannon Information
23(4)
References
23(4)
3 Probable and Improbable Data
27(14)
3.1 The Bayesian Interval
27(1)
3.2 Examples
28(5)
3.2.1 The Central Value of a Gaussian
28(1)
3.2.2 The Standard Deviation of a Gaussian
29(4)
3.3 Contour Lines
33(3)
3.4 On the Existence of the Bayesian Area
36(5)
References
38(3)
4 Description of Distributions I: Real x
41(14)
4.1 Gaussian Distributions
41(9)
4.1.1 The Simple Gaussian
41(4)
4.1.2 The Multidimensional Gaussian
45(2)
4.1.3 The Chi-Squared Model
47(3)
4.2 The Exponential Model
50(1)
4.3 Student's t-Distribution
51(4)
References
53(2)
5 Description of Distributions II: Natural x
55(8)
5.1 The Binomial Distribution
55(3)
5.2 The Multinomial Distribution
58(1)
5.3 The Poisson Distribution
59(4)
References
61(2)
6 Form Invariance I
63(18)
6.1 Groups
65(4)
6.2 The Symmetry of Form Invariance
69(2)
6.3 The Invariant Measure
71(2)
6.4 The Geometric Measure
73(1)
6.5 Form Invariance of the Posterior Distribution
74(1)
6.6 The Maximum Likelihood Estimator
75(2)
6.7 The Sufficient Statistic
77(1)
6.8 An Invariant Version of the Shannon Information
78(3)
References
79(2)
7 Examples of Invariant Measures
81(10)
7.1 Form Invariance Under Translations
81(1)
7.2 Form Invariance Under Dilations
82(1)
7.3 Form Invariance Under the Combination of Translation and Dilation
83(1)
7.4 A Rotational Invariance
84(3)
7.5 Special Triangular Matrices
87(1)
7.6 Triangular Matrices
87(4)
References
90(1)
8 A Linear Representation of Form Invariance
91(12)
8.1 A Vector Space of Functions
92(1)
8.2 An Orthogonal Transformation of the Function Space
93(2)
8.3 The Linear Representation of the Symmetry Groups
95(2)
8.4 The Linear Representation of Translation
97(1)
8.5 The Linear Representation of Dilation
98(2)
8.6 Linear Representation of Translation Combined with Dilation
100(3)
9 Going Beyond Form Invariance: The Geometric Prior
103(12)
9.1 Jeffreys' Rule
103(3)
9.2 Geometric Interpretation of the Prior μ
106(3)
9.3 On the Geometric Prior Distribution
109(1)
9.4 Examples of Geometric Priors
110(5)
9.4.1 An Expansion in Terms of Orthogonal Functions
110(1)
9.4.2 The Multinomial Model
111(2)
References
113(2)
10 Inferring the Mean or the Standard Deviation
115(12)
10.1 Inferring Both Parameters
115(6)
10.1.1 Multidimensional Gaussian Approximation to a Posterior Distribution
119(2)
10.2 Inferring the Mean Only
121(1)
10.3 Inferring the Standard Deviation Only
121(1)
10.4 The Argument of Neyman and Scott Against ML Estimation
122(5)
References
124(3)
11 Form Invariance II: Natural x
127(10)
11.1 The Binomial Model
128(3)
11.1.1 The Basic Binomial Model
128(1)
11.1.2 The Binomial Model for N Observations
129(2)
11.2 The Poisson Model as a Limit of the Binomial Model for >> n
131(6)
11.2.1 The Prior and Posterior of the Poisson Model
131(3)
11.2.2 The Poisson Model is Form Invariant
134(3)
12 Item Response Theory
137(14)
12.1 The Idea of and Some Notions Akin to Item Response Theory
138(2)
12.2 The Trigonometric Model of Item Response Theory
140(1)
12.3 Analysing Data with the Trigonometric Model
141(4)
12.3.1 The Guttman Scheme
141(1)
12.3.2 A Monte Carlo Game
142(3)
12.4 The Statistical Errors of the ML Estimators of Item Response Theory
145(6)
References
149(2)
13 On the Art of Fitting
151(16)
13.1 General Remarks
151(1)
13.2 The Chi-Squared Criterion
152(3)
13.3 A Histogram of Counts
155(5)
13.4 The Absolute Shannon Information
160(7)
13.4.1 Definition of the Absolute Shannon Information
160(1)
13.4.2 The Shannon Information Conveyed by a Chi-Squared Distribution
161(2)
13.4.3 Assigning Effective Numbers of Freedom
163(2)
References
165(2)
14 Summary
167(6)
14.1 The Starting Points of the Present Book
167(1)
14.2 Results
168(1)
14.3 Open Questions
169(4)
References
172(1)
Appendix A Problems and Solutions 173(34)
Appendix B Description of Distributions I: Real x 207(8)
Appendix C Form Invariance I 215(4)
Appendix D Beyond Form Invariance: The Geometric Prior 219(6)
Appendix E Inferring Mean or Standard Deviation 225(2)
Appendix F Form Invariance II: Natural x 227(2)
Appendix G Item Response Theory 229(6)
Appendix H On the Art of Fitting 235(6)
Index 241
Hanns Ludwig Harney, born in 1939, professor at the University of Heidelberg. He has contributed to experimental and theoretical physics within the Max-Planck Institute for Nuclear Physics at Heidelberg. His interest is focused on symmetries, such as isospin and its violation, as well as chaos, observed as reproducible fluctuations. Since the 1990's, the symmetry properties of common probability distributions lead him to a reformulation of Bayesian inference.
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