Markov chains and mixing times:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Weitere Verfasser: | , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2009]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvii, 371 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821847398 |
Internformat
MARC
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| 008 | 090213s2009 a||| |||| 00||| eng d | ||
| 010 | |a 2008031811 | ||
| 020 | |a 9780821847398 |9 978-0-8218-4739-8 | ||
| 035 | |a (OCoLC)612749095 | ||
| 035 | |a (DE-599)GBV572341067 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
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| 100 | 1 | |a Levin, David Asher |d 1971- |0 (DE-588)138179263 |4 aut | |
| 245 | 1 | 0 | |a Markov chains and mixing times |c David A. Levin ; Yuval Peres ; Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xvii, 371 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 505 | 8 | 0 | |t Coupling from the past (Chapter 22) |r Propp, James G. |t Coupling from the past (Chapter 22) |r Wilson, David B. |
| 650 | 0 | 7 | |a Markov-Kette |0 (DE-588)4037612-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Markov-Kette |0 (DE-588)4037612-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Peres, Yuval |0 (DE-588)121404420 |4 aut | |
| 700 | 1 | |a Wilmer, Elizabeth L. |d 1970- |0 (DE-588)137354932 |4 aut | |
| 700 | 1 | |a Propp, James G. |d 1960- |0 (DE-588)1115743872 |4 ctb | |
| 700 | 1 | |a Wilson, David B. |4 ctb | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1204-3 |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017115466 | |
Datensatz im Suchindex
| _version_ | 1810452473418088448 |
|---|---|
| adam_text |
Contents
Preface
xi
Overview
xii
For the Reader
xiii
For the Instructor
xiv
For the Expert
xvi
Acknowledgements
xvii
Part I: Basic Methods and Examples
1
Chapter
1.
Introduction to Finite Markov Chains
3
1.1.
Finite Markov Chains
3
1.2.
Random Mapping Representation
6
1.3.
Irreducibility and Aperiodicity
8
1.4.
Random Walks on Graphs
9
1.5.
Stationary Distributions
10
1.6.
Reversibility and Time Reversals
14
1.7.
Classifying the States of a Markov Chain*
16
Exercises
18
Notes
20
Chapter
2.
Classical (and Useful) Markov Chains
21
2.1.
Gambler's Ruin
21
2.2.
Coupon Collecting
22
2.3.
The Hypercube and the
Ehrenfest
Urn Model
23
2.4.
The
Pólya
Urn Model
25
2.5.
Birth-and-Death Chains
26
2.6.
Random Walks on Groups
27
2.7.
Random Walks on
Z
and Reflection Principles
30
Exercises
34
Notes
35
Chapter
3.
Markov Chain Monte Carlo: Metropolis and Glauber Chains
37
3.1.
Introduction
37
3.2.
Metropolis Chains
37
3.3.
Glauber Dynamics
40
Exercises
44
Notes
44
Chapter
4.
Introduction to Markov Chain Mixing
47
4.1.
Total Variation Distance
47
4.2.
Coupling and Total Variation Distance
49
4.3.
The Convergence Theorem
52
4.4.
Standardizing Distance from Stationarity
53
4.5.
Mixing Time
55
4.6.
Mixing and Time Reversal
55
4.7.
Ergodic Theorem*
58
Exercises
59
Notes
60
Chapter
5.
Coupling
63
5.1.
Definition
63
5.2.
Bounding Total Variation Distance
64
5.3.
Examples
65
5.4.
Grand Couplings
70
Exercises
73
Notes
74
Chapter
6.
Strong Stationary Times
75
6.1.
Top-to-Random Shuffle
75
6.2.
Definitions
76
6.3.
Achieving Equilibrium
77
6.4.
Strong Stationary Times and Bounding Distance
78
6.5.
Examples
80
6.6.
Stationary Times and Cesaro Mixing Time*
83
Exercises
84
Notes
85
Chapter
7.
Lower Bounds on Mixing Times
87
7.1.
Counting and Diameter Bounds
87
7.2.
Bottleneck Ratio
88
7.3.
Distinguishing Statistics
92
7.4.
Examples
96
Exercises
98
Notes
98
Chapter
8.
The Symmetric Group and Shuffling Cards
99
8.1.
The Symmetric Group
99
8.2.
Random Transpositions
101
8.3.
Riffle Shuffles
106
Exercises
109
Notes 111
Chapter
9.
Random Walks on Networks
115
9.1.
Networks and Reversible Markov Chains
115
9.2.
Harmonic Functions
116
9.3.
Voltages and Current Flows
117
9.4.
Effective Resistance
118
9.5.
Escape Probabilities on a Square
123
Exercises
124
Notes
125
Chapter
10.
Hitting Times
127
10.1.
Definition
127
10.2.
Random Target Times
128
10.3.
Commute Time
130
10.4.
Hitting Times for the Torus
133
10.5.
Bounding Mixing Times via Hitting Times
134
10.6.
Mixing for the Walk on Two Glued Graphs
138
Exercises
139
Notes
141
Chapter
11.
Cover Times
143
11.1.
Cover Times
143
11.2.
The Matthews Method
143
11.3.
Applications of the Matthews Method
147
Exercises
151
Notes
152
Chapter
12.
Eigenvalues
153
12.1.
The Spectral Representation of a Reversible Transition Matrix
153
12.2.
The Relaxation Time
154
12.3.
Eigenvalues and Eigenfunctions of Some Simple Random Walks
156
12.4.
Product Chains
160
12.5.
An
Ѓ
Bound
163
12.6.
Time Averages
165
Exercises
167
Notes
168
Part II: The Plot Thickens
169
Chapter
13.
Eigenfunctions and Comparison of Chains
171
13.1.
Bounds on Spectral Gap via Contractions
171
13.2.
Wilson's Method for Lower Bounds
172
13.3.
The Dirichlet Form and the Bottleneck Ratio
175
13.4.
Simple Comparison of Markov Chains
179
13.5.
The Path Method
182
13.6.
Expander Graphs*
185
Exercises
187
Notes
187
Chapter
14.
The Transportation Metric and Path Coupling
189
14.1.
The Transportation Metric
189
14.2.
Path Coupling
191
14.3.
Fast Mixing for Colorings
193
14.4.
Approximate Counting
195
Exercises
198
Notes
199
Chapter
15.
The Ising Model
201
15.1.
Fast Mixing at High Temperature
201
15.2.
The Complete Graph
203
15.3.
The Cycle
204
15.4.
The Tree
206
15.5.
Block Dynamics
208
15.6.
Lower Bound for Ising on Square*
211
Exercises
213
Notes
214
Chapter
16.
Prom Shuffling Cards to Shuffling Genes
217
16.1.
Random Adjacent Transpositions
217
16.2.
Shuffling Genes
221
Exercise
226
Notes
227
Chapter
17.
Martingales and Evolving Sets
229
17.1.
Definition and Examples
- 229
17.2.
Optional Stopping Theorem
231
17.3.
Applications
233
17.4.
Evolving Sets
235
17.5.
A General Bound on Return Probabilities
239
17.6.
Harmonic Functions and the Doob h-Transform
241
17.7.
Strong Stationary Times from Evolving Sets
243
Exercises
245
Notes
245
Chapter
18.
The Cutoff Phenomenon
247
18.1.
Definition
247
18.2.
Examples of Cutoff
248
18.3.
A Necessary Condition for Cutoff
252
18.4.
Separation Cutoff
254
Exercise
255
Notes
255
Chapter
19.
Lamplighter Walks
257
19.1.
Introduction
257
19.2.
Relaxation Time Bounds
258
19.3.
Mixing Time Bounds
260
19.4.
Examples
262
Notes
263
Chapter
20.
Continuous-Time Chains*
265
20.1.
Definitions
265
20.2.
Continuous-Time Mixing
266
20.3.
Spectral Gap
268
20.4.
Product Chains
269
Exercises
273
Notes
273
Chapter
21.
Countable State Space Chains*
275
21.1.
Recurrence and Transience
275
21.2.
Infinite Networks
277
21.3. Positive
Recurrence and Convergence
279
21.4.
Null Recurrence and Convergence
283
21.5.
Bounds on Return Probabilities
284
Exercises
285
Notes
286
Chapter
22.
Coupling from the Past
287
22.1.
Introduction
287
22.2.
Monotone CFTP
288
22.3.
Perfect Sampling via Coupling from the Past
293
22.4.
The Hardcore Model
294
22.5.
Random State of an Unknown Markov Chain
296
Exercise
297
Notes
297
Chapter
23.
Open Problems
299
23.1.
The Ising Model
299
23.2.
Cutoff
300
23.3.
Other Problems
301
Appendix A. Background Material
303
A.I. Probability Spaces and Random Variables
303
A.
2.
Metric Spaces
308
A.3. Linear Algebra
308
A.
4.
Miscellaneous
309
Appendix B. Introduction to Simulation
311
B.I. What Is Simulation?
311
B.2.
Von
Neumann Unbiasing*
312
B.3. Simulating Discrete Distributions and Sampling
313
B.4. Inverse Distribution Function Method
314
B.5. Acceptance-Rejection Sampling
314
B.6. Simulating Normal Random Variables
317
B.7. Sampling from the Simplex
318
B.8. About Random Numbers
318
B.9. Sampling from Large Sets*
319
Exercises
322
Notes
325
Appendix C. Solutions to Selected Exercises
327
Bibliography
353
Notation Index
363
Index
365 |
| any_adam_object | 1 |
| author | Levin, David Asher 1971- Peres, Yuval Wilmer, Elizabeth L. 1970- |
| author2 | Propp, James G. 1960- Wilson, David B. |
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| contents | Coupling from the past (Chapter 22) |
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| id | DE-604.BV035310702 |
| illustrated | Illustrated |
| indexdate | 2024-09-17T14:07:04Z |
| institution | BVB |
| isbn | 9780821847398 |
| language | English |
| lccn | 2008031811 |
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| spelling | Levin, David Asher 1971- (DE-588)138179263 aut Markov chains and mixing times David A. Levin ; Yuval Peres ; Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson Providence, Rhode Island American Mathematical Society [2009] © 2009 xvii, 371 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Coupling from the past (Chapter 22) Propp, James G. Coupling from the past (Chapter 22) Wilson, David B. Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s DE-604 Peres, Yuval (DE-588)121404420 aut Wilmer, Elizabeth L. 1970- (DE-588)137354932 aut Propp, James G. 1960- (DE-588)1115743872 ctb Wilson, David B. ctb Erscheint auch als Online-Ausgabe 978-1-4704-1204-3 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levin, David Asher 1971- Peres, Yuval Wilmer, Elizabeth L. 1970- Markov chains and mixing times Coupling from the past (Chapter 22) Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4037612-6 |
| title | Markov chains and mixing times |
| title_alt | Coupling from the past (Chapter 22) |
| title_auth | Markov chains and mixing times |
| title_exact_search | Markov chains and mixing times |
| title_full | Markov chains and mixing times David A. Levin ; Yuval Peres ; Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson |
| title_fullStr | Markov chains and mixing times David A. Levin ; Yuval Peres ; Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson |
| title_full_unstemmed | Markov chains and mixing times David A. Levin ; Yuval Peres ; Elizabeth L. Wilmer, With a chapter on "Coupling from the past" by James G. Propp and David B. Wilson |
| title_short | Markov chains and mixing times |
| title_sort | markov chains and mixing times |
| topic | Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Markov-Kette |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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