Introduction to Markov chains: with special emphasis on rapid mixing
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Braunschweig ; Wiesbaden
Vieweg
[2000]
|
| Schriftenreihe: | Advanced lectures in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | ix, 232 Seiten Illustrationen |
| ISBN: | 9783528069865 3528069864 |
Internformat
MARC
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| 245 | 1 | 0 | |a Introduction to Markov chains |b with special emphasis on rapid mixing |c Ehrhard Behrends |
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| 264 | 4 | |c © 2000 | |
| 300 | |a ix, 232 Seiten |b Illustrationen | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
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Contents
Part I Finite Markov chains (the background)
1
1
Markov chains: how to start?
4
Random walk, Markov chain, stochastic process, Markov process, Kolmogorov s
theorem, Maxkov chains vs. Markov processes, exercises.
2
Examples of Markov chains
12
Classical examples, Markov chains as graphs, shuffling, processes with a short
memory, exercises.
3
How linear algebra comes into play
19
fe-step transitions and the fc th power of the transition matrix, equilibrium
distribution, the loss-of-memory phenomenon, exercises.
4
The fundamental notions in connection with Markov chains
23
Closed subsets of a chain, irreducible chains, states which communicate, the
period of a state, recurrent and transient states, exercises.
5
Transient states
36
Fundamental matrix, running time until absorption, exercises.
6
An analytical lemma
42
The fundamental lemma of renewal theory, an example from renewal theory,
exercises.
7
Irreducible Markov chains
48
All states in an irreducible chain are positive recurrent, the equilibrium dis¬
tribution, irreducible chains with an arbitrary period, the first passage time
matrix, exercises.
8
Notes and remarks
61
Summary of part I, history, Markov chains on countable state spaces and in
continuous time, the ergodic theorem, exercises.
Part II Rapidly mixing chains
67
9
Perron-Frobenius theory
71
Eigenvalues and eigenvectors of a stochastic matrix, the canonical form, exer¬
cises.
10
Rapid mixing: a first approach
77
The time reversal chain, reversible chains, an estimate of the mixing rate using
eigenvalues, estimates which use the entries of the transition matrix directly,
applications to renewal theory, exercises.
viii
____________________________________________________________Contents
11
Conductance
91
Capacity, ergodic flow, conductance, chains defined by graphs, edge magnifi¬
cation, the method of canonical paths, applications, exercises.
12
Stopping times and the strong Markov property
102
Partial information vs.
sub-ťr-algebras, filtrations,
stopping times, the strong
Markov property, exercises.
13
Coupling methods
108
The total variation distance, examples of couplings, the coupling inequality,
coupled Markov chains, applications, the power of coupling methods, exercises.
14
Strong uniform times
121
Stopping times which are compatible with the Markov property, strong uniform
times, an application to shufflings, exercises.
15
Markov chains on finite groups I (commutative groups)
128
Probability measures on groups give rise to Markov chains, the character group
of a finite group, Fourier transform of functions and measures, the Fourier
inversion formula, the Plancherel formula, convolutions, characterization of
irreducible and aperiodic chains, rapid mixing on commutative groups, appli¬
cations, exercises.
16
Markov chains on finite groups II (arbitrary groups)
142
(Unitary) representations, irreducible representations, Schur s lemma, connect¬
ing matrices, duals of a group, properties of the coordinate functions, the
Peter-Weyl theorem, examples of duals, Fourier transform of functions and
measures, the Fourier inversion formula, the Plancherel formula, convolutions,
characterization of irreducible and aperiodic chains, rapid mixing on arbirary
groups, estimates which use matrix norms, selfadjointness or class measures,
applications, exercises.
17
Notes and remarks
168
Notes and remarks concerning part II, coupling from the past, exact simula¬
tion, cut-off-phenomenon.
Part III Rapidly mixing chains: applications
171
18
Random generation and counting
174
Self-reducible problems, the solution tree, counting implies simulation and vice
versa, simulation without counting, exercises.
19
Markov random fields
183
Random fields, neighbourhood systems, Markov random fields, local charac¬
teristics, the chain associated with a family of local characteristics, a coun¬
terexample, exercises.
20
Potentials, Gibbs fields, and the Ising model
195
Gibbs potentials and Gibbs fields, Gibbs fields are Markov random fields, the
toric Ising model, calculation of the partition function (the transfer matrix
method), Gibbs sampler, exercises.
їх
21
The Metropolis sampler and simulated annealing
209
Proposal chain, Metropolis sampler, the spectral gap of the proposal chain vs.
the gap of *-ne Metropolis chain, simulated annealing, a logarithmic cooling
schedule, exercises.
22
Notes and remarks
220
and remarks cone
,
image analysis.
tes
and remarks
Notes and remarks concerning part III, Monte-Carlo integration, volume esti-
matinn imaere analvsis.
Bibliography
223
Index
229
|
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| discipline | Mathematik |
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| id | DE-604.BV013012918 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:37:42Z |
| institution | BVB |
| isbn | 9783528069865 3528069864 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008867460 |
| oclc_num | 43695402 |
| open_access_boolean | |
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| physical | ix, 232 Seiten Illustrationen |
| publishDate | 2000 |
| publishDateSearch | 2000 |
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| publisher | Vieweg |
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| series2 | Advanced lectures in mathematics |
| spelling | Behrends, Ehrhard 1946- (DE-588)124668631 aut Introduction to Markov chains with special emphasis on rapid mixing Ehrhard Behrends Braunschweig ; Wiesbaden Vieweg [2000] © 2000 ix, 232 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Advanced lectures in mathematics Markov, Processus de Markov-processen gtt Markov processes Markov-Kette (DE-588)4037612-6 gnd rswk-swf Markov-Kette (DE-588)4037612-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-32290-157-6 Digitalisierung SABAschaffenburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008867460&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Behrends, Ehrhard 1946- Introduction to Markov chains with special emphasis on rapid mixing Markov, Processus de Markov-processen gtt Markov processes Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4037612-6 |
| title | Introduction to Markov chains with special emphasis on rapid mixing |
| title_auth | Introduction to Markov chains with special emphasis on rapid mixing |
| title_exact_search | Introduction to Markov chains with special emphasis on rapid mixing |
| title_full | Introduction to Markov chains with special emphasis on rapid mixing Ehrhard Behrends |
| title_fullStr | Introduction to Markov chains with special emphasis on rapid mixing Ehrhard Behrends |
| title_full_unstemmed | Introduction to Markov chains with special emphasis on rapid mixing Ehrhard Behrends |
| title_short | Introduction to Markov chains |
| title_sort | introduction to markov chains with special emphasis on rapid mixing |
| title_sub | with special emphasis on rapid mixing |
| topic | Markov, Processus de Markov-processen gtt Markov processes Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Markov, Processus de Markov-processen Markov processes Markov-Kette |
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