9781584881582

Complex Stochastic Systems

by ;
  • ISBN13:

    9781584881582

  • ISBN10:

    1584881585

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2000-08-09
  • Publisher: Chapman & Hall/

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Summary

Complex stochastic systems comprises a vast area of research, from modelling specific applications to model fitting, estimation procedures, and computing issues. The exponential growth in computing power over the last two decades has revolutionized statistical analysis and led to rapid developments and great progress in this emerging field. In Complex Stochastic Systems, leading researchers address various statistical aspects of the field, illustrated by some very concrete applications.A Primer on Markov Chain Monte Carlo by Peter J. Green provides a wide-ranging mixture of the mathematical and statistical ideas, enriched with concrete examples and more than 100 references.Causal Inference from Graphical Models by Steffen L. Lauritzen explores causal concepts in connection with modelling complex stochastic systems, with focus on the effect of interventions in a given system.State Space and Hidden Markov Models by Hans R. Künschshows the variety of applications of this concept to time series in engineering, biology, finance, and geophysics.Monte Carlo Methods on Genetic Structures by Elizabeth A. Thompson investigates special complex systems and gives a concise introduction to the relevant biological methodology.Renormalization of Interacting Diffusions by Frank den Hollander presents recent results on the large space-time behavior of infinite systems of interacting diffusions.Stein's Method for Epidemic Processes by Gesine Reinert investigates the mean field behavior of a general stochastic epidemic with explicit bounds.Individually, these articles provide authoritative, tutorial-style exposition and recent results from various subjects related to complex stochastic systems. Collectively, they link these separate areas of study to form the first comprehensive overview of this rapidly developing field.

Table of Contents

Contributors vii
Participants ix
Preface xiii
A Primer on Markov Chain Monte Carlo
1(62)
Peter J. Green
Introduction
1(1)
Getting started: Bayesian inference and the Gibbs sampler
2(9)
Bayes theorem and inference
2(1)
Cyclones example: point processes and change points
2(1)
constant rate
3(1)
constant rate, the Bayesian way
4(1)
The Gibbs sampler for a Normal random sample
5(2)
Cyclones example, continued
7(1)
constant rate, with hyperparameter
8(2)
constant rate, with change point
10(1)
multiple change points
10(1)
Other approaches to Bayesian computation
11(1)
MCMC --- the general idea and the main limit theorems
11(3)
The basic limit theorems
13(1)
Harris recurrence
13(1)
Rates of convergence
14(1)
Recipes for constructing MCMC methods
14(7)
The Gibbs sampler
15(1)
The Metropolis method
16(1)
The Metropolis-Hastings sampler
16(1)
Proof of detailed balance
16(1)
Updating several variables at once
17(1)
The role of the full conditionals
17(1)
Combining kernels to make an ergodic sampler
17(1)
Common choices for proposal distribution
18(1)
Comparing Metropolis-Hastings to rejection sampling
19(1)
Example: Weibull/Gamma experiment
19(1)
Cyclones example, continued
20(1)
another hyperparameter
20(1)
unknown change points
20(1)
The role of graphical models
21(4)
Directed acyclic graphs
21(2)
Undirected graphs, and spatial modelling
23(1)
Markov properties
23(2)
Modelling directly with an undirected graph
25(1)
Chain graphs
25(1)
Performance of MCMC methods
25(5)
Monitoring convergence
26(1)
Monte Carlo standard errors
27(1)
Blocking (or batching)
28(1)
Using empirical covariances
28(1)
Initial series estimators
29(1)
Regeneration
29(1)
Regeneration using Nummelin's splitting
29(1)
Reversible jump methods
30(12)
Explicit representation using random numbers
32(1)
MCMC for variable dimension problems
32(2)
Example: step functions
34(2)
Cyclones example, continued
36(1)
unknown number of change points
36(1)
with a cyclic component
37(1)
Bayesian model determination
38(1)
Within-model simulation
39(2)
Estimating the marginal likelihood
41(1)
Across-model simulation
41(1)
Some tools for improving performance
42(6)
Tuning a MCMC simulation
42(1)
Antithetic variables and over-relaxation
42(1)
Augmenting the state space
43(1)
Simulated tempering
44(1)
Simulated tempering, by changing the temperature
44(1)
Simulated tempering, by inventing models
45(1)
Auxiliary variables
46(2)
Coupling from the Past (CFTP)
48(3)
Is CFTP of any use in statistics?
49(1)
The Rejection Coupler
49(2)
Towards generic methods for Bayesian statistics
51(1)
Miscellaneous topics
51(3)
Diffusion methods
51(1)
Sensitivity analysis via MCMC
52(1)
Bayes with a loss function
53(1)
Some notes on programming MCMC
54(2)
The Bugs software
54(1)
Your own code
54(1)
High and low level languages
55(1)
Validating your code
55(1)
Conclusions
56(1)
Some strengths of MCMC
56(1)
Some weaknesses and dangers
56(1)
Some important lines of continuing research
57(1)
References and further reading
57(6)
Causal Inference from Graphical Models
63(46)
Steffen L. Lauritzen
Introduction
63(1)
Graph terminology
64(2)
Conditional independence
66(3)
Markov properties for undirected graphs
69(2)
The directed Markov property
71(4)
Causal Markov models
75(4)
Conditioning by observation or intervention
75(1)
Causal graphs
76(3)
Assessment of treatment effects in sequential trials
79(2)
Identifiability of causal effects
81(14)
The general problem
81(1)
Intervention graphs
82(1)
Three inference rules
83(1)
The back-door formulae
84(2)
Confounding
86(1)
Randomization
86(1)
Sufficient covariate
87(1)
Partial compliance
87(1)
The front-door formula
88(2)
Additional examples
90(5)
Structural equation models
95(2)
Potential responses and counterfactuals
97(3)
Partial compliance revisited
98(2)
Other issues
100(2)
Extension to chain graphs
100(1)
Causal discovery
101(1)
References
102(7)
State Space and Hidden Markov Models
109(66)
Hans R. Kunsch
Introduction
109(1)
The general state space model
110(11)
Examples
111(1)
ARMA models as state space models
112(2)
Structural time series models
114(1)
Engineering examples
115(1)
Biological examples
116(2)
Examples from finance mathematics
118(1)
Geophysical examples
119(1)
General remarks on model building
120(1)
Filtering and smoothing recursions
121(12)
Filtering
121(1)
Smoothing
122(3)
Posterior mode and dynamic programming
125(1)
The reference probability method
126(1)
Transitions that are not absolutely continuous
127(1)
Forgetting of the initial distribution
128(5)
Exact and approximate filtering and smoothing
133(4)
Hidden Markov models
133(1)
Kalman filter
134(1)
The innovation form of state space models
135(1)
Exact computations in other cases
135(1)
Extended Kalman filter
136(1)
Other approximate methods
136(1)
Monte Carlo filtering and smoothing
137(14)
Markov chain Monte Carlo: single updates
137(1)
Markov chain Monte Carlo: multiple updates
138(1)
Recursive Monte Carlo filtering
139(3)
Recursive Monte Carlo smoothing
142(1)
Examples
143(4)
Error propagation in the recursive Monte Carlo filter
147(4)
Parameter estimation
151(9)
Bayesian methods
151(1)
Monte Carlo likelihood approximations based on prediction samples
152(2)
Monte Carlo approximations based on smoother samples
154(1)
Examples
155(4)
Asymptotics of the maximum likelihood estimator
159(1)
Extensions of the model
160(7)
Spatial models
160(4)
Stochastic context-free grammars
164(3)
References
167(8)
Monte Carlo Methods on Genetic Structures
175(44)
Elizabeth A. Thompson
Genetics, pedigrees, and structured systems
175(12)
Introduction to Genetics
175(2)
The conditional independence structures of genetics
177(5)
``Peeling'': sequential computation
182(2)
The Baum algorithm for conditional probabilities
184(3)
Computations on pedigrees
187(14)
Peeling meiosis indicators
187(1)
Peeling genotypes on pedigrees
188(3)
Importance sampling and Monte Carlo likelihood
191(3)
Risk, Elods, conditional Elods, and sequential imputation
194(1)
Risk probabilities
194(1)
Elods
195(1)
Elods conditional on trait data
196(1)
Monte Carlo likelihood by sequential imputation
196(2)
Monte Carlo likelihood ratio estimation
198(1)
Monte Carlo likelihood ratios
198(1)
Monte Carlo likelihood surfaces
199(2)
MCMC methods for Multilocus Genetic Data
201(13)
Monte Carlo estimation of location score curves
201(3)
Markov chain Monte Carlo
204(3)
Single-site updating methods
207(3)
Block updating; combining exact and MC computation
210(2)
Tightly-linked loci: the M-sampler
212(2)
Conclusion
214(1)
References
215(4)
Renormalization of Interacting Diffusions
219(16)
Frank den Hollander
Introduction
219(1)
The model
220(1)
Interpretation of the model
221(1)
Block averages and renormalization
221(1)
The hierarchical lattice
222(3)
The renormalization transformation
225(2)
Analysis of the orbit
227(3)
Higher-dimensional state spaces
230(1)
Open problems
231(1)
Conclusion
232(1)
References
232(3)
Stein's Method for Epidemic Processes
235(42)
Gesine Reinert
Introduction
235(4)
A brief introduction to Stein's method
239(8)
Stein's method for normal approximations
239(4)
Stein's method in general
243(1)
Stein's method for the weak law of large numbers
244(1)
Stein's method for the weak law in measure space
245(2)
The distance of the GSE to its mean field limit
247(22)
Assumptions
247(2)
Heuristics
249(1)
Previous results
250(4)
A bound on the distance to the mean field limit
254(10)
A special case: λ(t, x) = αx(t)
264(4)
Some plots of the limiting expression
268(1)
Discussion
269(4)
References
273(4)
Index 277

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