Introduction to Bayesian statistics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English German |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2007
|
| Ausgabe: | 2., updated and enl. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | 1. Aufl. u.d.T.: Koch, Karl-Rudolf: Einführung in die Bayes-Statistik |
| Beschreibung: | XII, 249 S. graph. Darst. |
| ISBN: | 9783540727231 354072723X |
Internformat
MARC
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| 020 | |a 9783540727231 |c Gb. : ca. EUR 96.25 (freier Pr.), ca. sfr 147.50 (freier Pr.) |9 978-3-540-72723-1 | ||
| 020 | |a 354072723X |c Gb. : ca. EUR 96.25 (freier Pr.), ca. sfr 147.50 (freier Pr.) |9 3-540-72723-X | ||
| 024 | 3 | |a 9783540727231 | |
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| 100 | 1 | |a Koch, Karl-Rudolf |d 1935- |e Verfasser |0 (DE-588)121013820 |4 aut | |
| 245 | 1 | 0 | |a Introduction to Bayesian statistics |c Karl-Rudolf Koch |
| 250 | |a 2., updated and enl. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XII, 249 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a 1. Aufl. u.d.T.: Koch, Karl-Rudolf: Einführung in die Bayes-Statistik | ||
| 650 | 7 | |a Methode van Bayes |2 gtt | |
| 650 | 4 | |a Statistique bayésienne | |
| 650 | 4 | |a Bayesian statistical decision theory | |
| 650 | 0 | 7 | |a Bayes-Entscheidungstheorie |0 (DE-588)4144220-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Bayes-Verfahren |0 (DE-588)4204326-8 |2 gnd |9 rswk-swf |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739409&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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Datensatz im Suchindex
| _version_ | 1804136643244851200 |
|---|---|
| adam_text | Contents
1
Introduction
1
2
Probability
3
2.1
Rules of Probability
....................... 3
2.1.1
Deductive and Plausible Reasoning
........... 3
2.1.2
Statement Calculus
.................... 3
2.1.3
Conditional Probability
................. 5
2.1.4
Product Rule and Sum Rule of Probability
...... 6
2.1.5
Generalized Sum Rule
.................. 7
2.1.6
Axioms of Probability
.................. 9
2.1.7
Chain Rule and Independence
.............. 11
2.1.8
Bayes
Theorem
..................... 12
2.1.9
Recursive Application of
Bayes
Theorem
....... 16
2.2
Distributions
........................... 16
2.2.1
Discrete Distribution
................... 17
2.2.2
Continuous Distribution
................. 18
2.2.3
Binomial Distribution
.................. 20
2.2.4
Multidimensional Discrete and Continuous Distributions
22
2.2.5
Marginal Distribution
.................. 24
2.2.6
Conditional Distribution
................. 26
2.2.7
Independent Random Variables and Chain Rule
... 28
2.2.8
Generalized
Bayes
Theorem
.............. 31
2.3
Expected Value, Variance and Covariance
........... 37
2.3.1
Expected Value
...................... 37
2.3.2
Variance and Covariance
................. 41
2.3.3
Expected Value of a Quadratic Form
.......... 44
2.4
Univariate Distributions
..................... 45
2.4.1
Normal Distribution
................... 45
2.4.2
Gamma Distribution
................... 47
2.4.3
Inverted Gamma Distribution
.............. 48
2.4.4
Beta Distribution
..................... 48
2.4.5
x2-Distribution
...................... 48
2.4.6
F-Distribution
...................... 49
2.4.7
¿-Distribution
....................... 49
2.4.8
Exponential Distribution
................ 50
2.4.9
Cauchy Distribution
................... 51
2.5
Multivariate Distributions
.................... 51
2.5.1
Multivariate Normal Distribution
............ 51
2.5.2
Multivariate i-Distribution
............... 53
X
Contents
2.5.3
Normal-Gamma
Distribution.............. 55
2.6
Prior
Density Functions
..................... 56
2.6.1
Noninformative Priors
.................. 56
2.6.2
Maximum Entropy Priors
................ 57
2.6.3
Conjugate Priors
..................... 59
3
Parameter Estimation, Confidence Regions and Hypothesis
Testing
63
3.1
Bayes
Rule
............................ 63
3.2
Point Estimation
......................... 65
3.2.1
Quadratic Loss Function
................. 65
3.2.2
Loss Function of the Absolute Errors
.......... 67
3.2.3
Zero-One Loss
...................... 69
3.3
Estimation of Confidence Regions
................ 71
3.3.1
Confidence Regions
.................... 71
3.3.2
Boundary of a Confidence Region
............ 73
3.4
Hypothesis Testing
........................ 73
3.4.1
Different Hypotheses
................... 74
3.4.2
Test of Hypotheses
.................... 75
3.4.3
Special Priors for Hypotheses
.............. 78
3.4.4
Test of the Point Null Hypothesis by Confidence Regions
82
4
Linear Model
85
4.1
Definition and Likelihood Function
............... 85
4.2
Linear Model with Known Variance Factor
.......... 89
4.2.1
Noninformative Priors
.................. 89
4.2.2
Method of Least Squares
................ 93
4.2.3
Estimation of the Variance Factor in Traditional
Statistics
......................... 94
4.2.4
Linear Model with Constraints in Traditional
Statistics
......................... 96
4.2.5
Robust Parameter Estimation
.............. 99
4.2.6
Informative Priors
.................... 103
4.2.7
Kalman
Filter
....................... 107
4.3
Linear Model with Unknown Variance Factor
......... 110
4.3.1
Noninformative Priors
.................. 110
4.3.2
Informative Priors
.................... 117
4.4
Linear Model not of Full Rank
................. 121
4.4.1
Noninformative Priors
.................. 122
4.4.2
Informative Priors
.................... 124
5
Special Models and Applications
129
5.1
Prediction and Filtering
..................... 129
5.1.1
Model of Prediction and Filtering as Special Linear
Model
........................... 130
Contents
XI
5.1.2 Special Model
of Prediction and Filtering
....... 135
5.2
Variance and Covariance Components
............. 139
5.2.1
Model and Likelihood Function
............. 139
5.2.2
Noninformative
Priors
.................. 143
5.2.3
Informative Priors
.................... 143
5.2.4
Variance Components
.................. 144
5.2.5
Distributions for Variance Components
........ 148
5.2.6
Regularization
...................... 150
5.3
Reconstructing and Smoothing of Three-dimensional Images
. 154
5.3.1
Positron Emission Tomography
............. 155
5.3.2
Image Reconstruction
.................. 156
5.3.3
Iterated Conditional Modes Algorithm
......... 158
5.4
Pattern Recognition
....................... 159
5.4.1
Classification by
Bayes
Rule
............... 160
5.4.2
Normal Distribution with Known and Unknown
Parameters
........................ 161
5.4.3
Parameters for Texture
................. 163
5.5
Bayesian Networks
........................ 167
5.5.1
Systems with Uncertainties
............... 167
5.5.2
Setup of a Bayesian Network
.............. 169
5.5.3
Computation of Probabilities
.............. 173
5.5.4
Bayesian Network in Form of a Chain
......... 181
5.5.5
Bayesian Network in Form of a Tree
.......... 184
5.5.6
Bayesian Network in Form of a Polytreee
....... 187
6
Numerical Methods
193
6.1
Generating Random Values
................... 193
6.1.1
Generating Random Numbers
.............. 193
6.1.2
Inversion Method
..................... 194
6.1.3
Rejection Method
.................... 196
6.1.4
Generating Values for Normally Distributed Random
Variables
......................... 197
6.2
Monte Carlo Integration
..................... 197
6.2.1
Importance Sampling and SIR Algorithm
....... 198
6.2.2
Crude Monte Carlo Integration
............. 201
6.2.3
Computation of Estimates, Confidence Regions and
Probabilities for Hypotheses
............... 202
6.2.4
Computation of Marginal Distributions
........ 204
6.2.5
Confidence Region for Robust Estimation of
Parameters as Example
................. 207
6.3
Markov Chain Monte Carlo Methods
.............. 216
6.3.1
Metropolis Algorithm
.................. 216
6.3.2
Gibbs Sampler
...................... 217
6.3.3
Computation of Estimates, Confidence Regions and
Probabilities for Hypotheses
............... 219
XII Contents
6.3.4
Computation of Marginal Distributions
........ 222
6.3.5
Gibbs Sampler for Computing and Propagating
Large Covariance Matrices
............... 224
6.3.6
Continuation of the Example: Confidence Region for
Robust Estimation of Parameters
............ 229
References
235
Index
245
Karl-Rudolf Koch
Introduction
to Bayesian Statistics
-
Second Edition
The Introduction
to
Bayesian
Statistics
(2nd
edition) presents
Bayes
theorem, the
estimation of unknown parameters, the determination of confidence regions and the
derivation of tests of hypotheses for the unknown parameters, in a manner that is
simple, intuitive and easy to comprehend. The methods are applied to linear models, in
models for a robust estimation, for prediction and filtering and in models for estimating
variance components and covariance components. Regularization of inverse problems
and pattern recognition are also covered while Bayesian networks serve for reaching deci¬
sions in systems with uncertainties. It analytical solutions cannot be derived, numerical
algorithms are presented, such as the Monte Carlo integration and Markov Chain Monte
Carlo methods.
Karl-Rudolf Koch, born
1935
in Hilchenbach, Germany, studied geodesy
from
1955
to
1959
at the University of Bonn. After receiving the degree
of a Dr.-Ing. he worked first as research associate at the Ohio State
University in Columbus, Ohio, USA, and then as research geodesist
at the National Geodetic Survey in Rockville, Maryland, USA. Having
been protessor for physical geodesy tor eight years
trom
1970
onwards,
he became professor for theoretical geodesy and director of the
Institute of Theoretical Geodesy of the University of Bonn in
1978.
He was member of the radar altimeter group of
ESA
from
1980-1987,
and then became director of the German Geodetic Research Institute
with departments in Munich and
Frankfurt/M.
until
1997.
In
1994
he received the honorary degree of a Dr.-Ing. from the Techni¬
cal University of Aachen and
1999
from the University of Stuttgart.
Since
2000
he is professor emeritus ot the University ot Bonn.
ISBN
978-3-540-72723-1
783540 727231|
>springer.com
|
| adam_txt |
Contents
1
Introduction
1
2
Probability
3
2.1
Rules of Probability
. 3
2.1.1
Deductive and Plausible Reasoning
. 3
2.1.2
Statement Calculus
. 3
2.1.3
Conditional Probability
. 5
2.1.4
Product Rule and Sum Rule of Probability
. 6
2.1.5
Generalized Sum Rule
. 7
2.1.6
Axioms of Probability
. 9
2.1.7
Chain Rule and Independence
. 11
2.1.8
Bayes'
Theorem
. 12
2.1.9
Recursive Application of
Bayes'
Theorem
. 16
2.2
Distributions
. 16
2.2.1
Discrete Distribution
. 17
2.2.2
Continuous Distribution
. 18
2.2.3
Binomial Distribution
. 20
2.2.4
Multidimensional Discrete and Continuous Distributions
22
2.2.5
Marginal Distribution
. 24
2.2.6
Conditional Distribution
. 26
2.2.7
Independent Random Variables and Chain Rule
. 28
2.2.8
Generalized
Bayes'
Theorem
. 31
2.3
Expected Value, Variance and Covariance
. 37
2.3.1
Expected Value
. 37
2.3.2
Variance and Covariance
. 41
2.3.3
Expected Value of a Quadratic Form
. 44
2.4
Univariate Distributions
. 45
2.4.1
Normal Distribution
. 45
2.4.2
Gamma Distribution
. 47
2.4.3
Inverted Gamma Distribution
. 48
2.4.4
Beta Distribution
. 48
2.4.5
x2-Distribution
. 48
2.4.6
F-Distribution
. 49
2.4.7
¿-Distribution
. 49
2.4.8
Exponential Distribution
. 50
2.4.9
Cauchy Distribution
. 51
2.5
Multivariate Distributions
. 51
2.5.1
Multivariate Normal Distribution
. 51
2.5.2
Multivariate i-Distribution
. 53
X
Contents
2.5.3
Normal-Gamma
Distribution. 55
2.6
Prior
Density Functions
. 56
2.6.1
Noninformative Priors
. 56
2.6.2
Maximum Entropy Priors
. 57
2.6.3
Conjugate Priors
. 59
3
Parameter Estimation, Confidence Regions and Hypothesis
Testing
63
3.1
Bayes
Rule
. 63
3.2
Point Estimation
. 65
3.2.1
Quadratic Loss Function
. 65
3.2.2
Loss Function of the Absolute Errors
. 67
3.2.3
Zero-One Loss
. 69
3.3
Estimation of Confidence Regions
. 71
3.3.1
Confidence Regions
. 71
3.3.2
Boundary of a Confidence Region
. 73
3.4
Hypothesis Testing
. 73
3.4.1
Different Hypotheses
. 74
3.4.2
Test of Hypotheses
. 75
3.4.3
Special Priors for Hypotheses
. 78
3.4.4
Test of the Point Null Hypothesis by Confidence Regions
82
4
Linear Model
85
4.1
Definition and Likelihood Function
. 85
4.2
Linear Model with Known Variance Factor
. 89
4.2.1
Noninformative Priors
. 89
4.2.2
Method of Least Squares
. 93
4.2.3
Estimation of the Variance Factor in Traditional
Statistics
. 94
4.2.4
Linear Model with Constraints in Traditional
Statistics
. 96
4.2.5
Robust Parameter Estimation
. 99
4.2.6
Informative Priors
. 103
4.2.7
Kalman
Filter
. 107
4.3
Linear Model with Unknown Variance Factor
. 110
4.3.1
Noninformative Priors
. 110
4.3.2
Informative Priors
. 117
4.4
Linear Model not of Full Rank
. 121
4.4.1
Noninformative Priors
. 122
4.4.2
Informative Priors
. 124
5
Special Models and Applications
129
5.1
Prediction and Filtering
. 129
5.1.1
Model of Prediction and Filtering as Special Linear
Model
. 130
Contents
XI
5.1.2 Special Model
of Prediction and Filtering
. 135
5.2
Variance and Covariance Components
. 139
5.2.1
Model and Likelihood Function
. 139
5.2.2
Noninformative
Priors
. 143
5.2.3
Informative Priors
. 143
5.2.4
Variance Components
. 144
5.2.5
Distributions for Variance Components
. 148
5.2.6
Regularization
. 150
5.3
Reconstructing and Smoothing of Three-dimensional Images
. 154
5.3.1
Positron Emission Tomography
. 155
5.3.2
Image Reconstruction
. 156
5.3.3
Iterated Conditional Modes Algorithm
. 158
5.4
Pattern Recognition
. 159
5.4.1
Classification by
Bayes
Rule
. 160
5.4.2
Normal Distribution with Known and Unknown
Parameters
. 161
5.4.3
Parameters for Texture
. 163
5.5
Bayesian Networks
. 167
5.5.1
Systems with Uncertainties
. 167
5.5.2
Setup of a Bayesian Network
. 169
5.5.3
Computation of Probabilities
. 173
5.5.4
Bayesian Network in Form of a Chain
. 181
5.5.5
Bayesian Network in Form of a Tree
. 184
5.5.6
Bayesian Network in Form of a Polytreee
. 187
6
Numerical Methods
193
6.1
Generating Random Values
. 193
6.1.1
Generating Random Numbers
. 193
6.1.2
Inversion Method
. 194
6.1.3
Rejection Method
. 196
6.1.4
Generating Values for Normally Distributed Random
Variables
. 197
6.2
Monte Carlo Integration
. 197
6.2.1
Importance Sampling and SIR Algorithm
. 198
6.2.2
Crude Monte Carlo Integration
. 201
6.2.3
Computation of Estimates, Confidence Regions and
Probabilities for Hypotheses
. 202
6.2.4
Computation of Marginal Distributions
. 204
6.2.5
Confidence Region for Robust Estimation of
Parameters as Example
. 207
6.3
Markov Chain Monte Carlo Methods
. 216
6.3.1
Metropolis Algorithm
. 216
6.3.2
Gibbs Sampler
. 217
6.3.3
Computation of Estimates, Confidence Regions and
Probabilities for Hypotheses
. 219
XII Contents
6.3.4
Computation of Marginal Distributions
. 222
6.3.5
Gibbs Sampler for Computing and Propagating
Large Covariance Matrices
. 224
6.3.6
Continuation of the Example: Confidence Region for
Robust Estimation of Parameters
. 229
References
235
Index
245
Karl-Rudolf Koch
Introduction
to Bayesian Statistics
-
Second Edition
The Introduction
to
Bayesian
Statistics
(2nd
edition) presents
Bayes'
theorem, the
estimation of unknown parameters, the determination of confidence regions and the
derivation of tests of hypotheses for the unknown parameters, in a manner that is
simple, intuitive and easy to comprehend. The methods are applied to linear models, in
models for a robust estimation, for prediction and filtering and in models for estimating
variance components and covariance components. Regularization of inverse problems
and pattern recognition are also covered while Bayesian networks serve for reaching deci¬
sions in systems with uncertainties. It analytical solutions cannot be derived, numerical
algorithms are presented, such as the Monte Carlo integration and Markov Chain Monte
Carlo methods.
Karl-Rudolf Koch, born
1935
in Hilchenbach, Germany, studied geodesy
from
1955
to
1959
at the University of Bonn. After receiving the degree
of a Dr.-Ing. he worked first as research associate at the Ohio State
University in Columbus, Ohio, USA, and then as research geodesist
at the National Geodetic Survey in Rockville, Maryland, USA. Having
been protessor for physical geodesy tor eight years
trom
1970
onwards,
he became professor for theoretical geodesy and director of the
Institute of Theoretical Geodesy of the University of Bonn in
1978.
He was member of the radar altimeter group of
ESA
from
1980-1987,
and then became director of the German Geodetic Research Institute
with departments in Munich and
Frankfurt/M.
until
1997.
In
1994
he received the honorary degree of a Dr.-Ing. from the Techni¬
cal University of Aachen and
1999
from the University of Stuttgart.
Since
2000
he is professor emeritus ot the University ot Bonn.
ISBN
978-3-540-72723-1
783540"727231|
>springer.com |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Koch, Karl-Rudolf 1935- |
| author_GND | (DE-588)121013820 |
| author_facet | Koch, Karl-Rudolf 1935- |
| author_role | aut |
| author_sort | Koch, Karl-Rudolf 1935- |
| author_variant | k r k krk |
| building | Verbundindex |
| bvnumber | BV022532849 |
| callnumber-first | Q - Science |
| callnumber-label | QA279 |
| callnumber-raw | QA279.5 |
| callnumber-search | QA279.5 |
| callnumber-sort | QA 3279.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 233 SK 830 |
| classification_tum | MAT 622f |
| ctrlnum | (OCoLC)163345879 (DE-599)DNB984419241 |
| dewey-full | 526.101519542 519.5/42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 526 - Mathematical geography 519 - Probabilities and applied mathematics |
| dewey-raw | 526.101519542 519.5/42 |
| dewey-search | 526.101519542 519.5/42 |
| dewey-sort | 3526.101519542 |
| dewey-tens | 520 - Astronomy and allied sciences 510 - Mathematics |
| discipline | Physik Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Physik Mathematik Wirtschaftswissenschaften |
| edition | 2., updated and enl. ed. |
| format | Book |
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| id | DE-604.BV022532849 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:07:16Z |
| indexdate | 2024-07-09T20:59:39Z |
| institution | BVB |
| isbn | 9783540727231 354072723X |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015739409 |
| oclc_num | 163345879 |
| open_access_boolean | |
| owner | DE-824 DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-945 DE-11 DE-703 |
| owner_facet | DE-824 DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-945 DE-11 DE-703 |
| physical | XII, 249 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| spelling | Koch, Karl-Rudolf 1935- Verfasser (DE-588)121013820 aut Introduction to Bayesian statistics Karl-Rudolf Koch 2., updated and enl. ed. Berlin [u.a.] Springer 2007 XII, 249 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier 1. Aufl. u.d.T.: Koch, Karl-Rudolf: Einführung in die Bayes-Statistik Methode van Bayes gtt Statistique bayésienne Bayesian statistical decision theory Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Bayes-Entscheidungstheorie (DE-588)4144220-9 s DE-604 Bayes-Verfahren (DE-588)4204326-8 s 1\p DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739409&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739409&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Koch, Karl-Rudolf 1935- Introduction to Bayesian statistics Methode van Bayes gtt Statistique bayésienne Bayesian statistical decision theory Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Bayes-Verfahren (DE-588)4204326-8 gnd |
| subject_GND | (DE-588)4144220-9 (DE-588)4204326-8 |
| title | Introduction to Bayesian statistics |
| title_auth | Introduction to Bayesian statistics |
| title_exact_search | Introduction to Bayesian statistics |
| title_exact_search_txtP | Introduction to Bayesian statistics |
| title_full | Introduction to Bayesian statistics Karl-Rudolf Koch |
| title_fullStr | Introduction to Bayesian statistics Karl-Rudolf Koch |
| title_full_unstemmed | Introduction to Bayesian statistics Karl-Rudolf Koch |
| title_short | Introduction to Bayesian statistics |
| title_sort | introduction to bayesian statistics |
| topic | Methode van Bayes gtt Statistique bayésienne Bayesian statistical decision theory Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Bayes-Verfahren (DE-588)4204326-8 gnd |
| topic_facet | Methode van Bayes Statistique bayésienne Bayesian statistical decision theory Bayes-Entscheidungstheorie Bayes-Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739409&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739409&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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