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Complex Stochastic Systems
Contents
List of Contributors
List of Participants
Preface
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CHAPTER 1: A Primer on Markov Chain Monte Carlo
1.1 Introduction
1.2 Getting started: Bayesian inference and the Gibbs sampler
1.2.1 Bayes theorem and inference
1.2.2 Cyclones example: point processes and change points
Model 1: constant rate
Model 2: constant rate, the Bayesian way
1.2.3 The Gibbs sampler for a Normal random sample
1.2.4 Cyclones example, continued
Model 3: constant rate, with hyperparameter
Model 4: constant rate, with change point
Model 5: multiple change points
1.2.5 Other approaches to Bayesian computation
1.3 MCMCโthe general idea and the main limit theorems
1.3.1 The basic limit theorems
1.3.2 Harris recurrence
1.3.3 Rates of convergence
1.4 Recipes for constructing MCMC methods
1.4.l The Gibbs sampler
1.4.2 The Metropolis method
1.4.3 The Metropolis-Hustings sampler
l.4.4 Proof of detailed balance
1.4.5 Updating several variables at once
1.4.6 The role of the full conditionals
1.4.7 Combining kernels to make an ergodic sampler
1.4.8 Common choices for proposal distribution
1.4.9 Comparing Metropolis-Hastings to rejection sampling
1.4.10 Example: Weibull/Gamma experiment
1.4.11 Cyclones example, continued
Model 6: another hyperparameter
Model 7: unknown change points
1.5 The role of graphical models
1.5.1 Directed acyclic graphs
1.5.2 Undirected graphs, and spatial modelling
Markov properties
Modelling directly with an undirected graph
1.5.3 Chain graphs
1.6 Performance of MCMC methods
1.6.1 Monitoring convergence
1.6.2 Monte Carlo standard errors
Blocking (or batching)
Using empirical covariances
Initial series estimators
Regeneration
Regeneration using Nummelinโs splitting
1.7 Reversible jump methods: Metropolis-Hastings in a more general setting
1.7.1 Explicit representation using random numbers
1.7.2 MCMC for variable dimension problems
1.7.3 Example: step functions
1.7.4 Cyclones example, continued
Model 8: unknown number of change points
Model 9: with a cyclic component
1.7.5 Bayesian model determination
Within-model simulation
Estimating the marginal likelihood
Across-model simulation
1.8 Some tools for improving performance
1.8.1 Tuning a MCMC simulation
1.8.2 Antithetic variables and over-relaxation
1.8.3 Augmenting the state space
1.8.4 Simulated tempering
Simulated tempering, by changing the temperature
Simulated tempering, by inventing models
1.8.5 Auxiliary variables
1.9 Coupling from the Past (CFTP)
1.9.1 Is CFTP of any use in statistics?
1.9.2 The Rejection Coupler
1.9.3 Towards generic methods for Bayesian statistics
1.10 Miscellaneous topics
1.10.1 Diffusion methods
1.10.2 Sensitivity analysis via MCMC
1.10.3 Bayes with a loss function
1.11 Some notes on programming MCMC
1.11.1 The BUGS software
1.11.2 Your own code
High and low level languages
Validating your code
1.12 Conclusions
1.12.1 Some strengths of MCMC
1.12.2 Some weaknesses and dangers
1.12.3 Some important lines of continuing research
Acknowledgements
1.13 References and further reading
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CHAPTER 2: Causal Inference from Graphical Models
2.1 Introduction
2.2 Graph terminology
2.3 Conditional independence
2.4 Markov properties for undirected graphs
2.5 The directed Markov property
2.6 Causal Markov models
2.6.1 Conditioning by observation or intervention
2.6.2 Causal graphs
2.7 Assessment of treatment effects in sequential trials
2.8 Identifiability of causal effects
2.8.1 The general problem
2.8.2 Intervention graphs
2.8.3 Three inference rules
2.8.4 The back-door formulae
Confounding
Randomization
Sufficient covariate
Partial compliance
2.8.5 The front-door formula
2.8.6 Additional examples
2.9 Structural equation models
2.10 Potential responses and counterfactuals
2.10.1 Partial compliance revisited
2.11 Other issues
2.11.1 Extension to chain graphs
2.11.2 Causal discovery
Acknowledgements
2.12 References
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CHAPTER 3: State Space and Hidden Markov Models
3.1 Introduction
3.2 The general state space model
3.2.1 Examples
ARMA models as state space models
Structural tame series models
Engineering examples
Biological examples
Examples from finance mathematics
Geophysical examples
3.2.2 General remarks on model building
3.3 Filtering and smoothing recursions
3.3.1 Filtering
3.3.2 Smoothing
3.3.3 Posterior mode and dynamic programming
3.3.4 The reference probability method
3.3.5 Transitions that are not absolutely continuous
3.3.6 Forgetting of the initial distribution
3.4 Exact and approximate methods for filtering and smoothing
3.4.1 Hidden Markov models
3.4.2 Kalman filter
3.4.3 The innovation form of state space models
3.4.4 Exact computations in other cases
3.4.5 Extended Kalman filter
3.4.6 Other approximate methods
3.5 Monte Carlo methods for filtering and smoothing
3.5.1 Markov chain Monte Carlo: single updates
3.5.2 Markov chain Monte Carlo: multiple updates
3.5.3 Recursive Monte Curlo filtering
3.5.4 Recursive Monte Curlo smoothing
3.5.5 Examples
3.5.6 Error propagation in the recursive Monte Carlo filter
3.6 Parameter estimation
3.6.1 Bayesian methods
3.6.2 Monte Carlo likelihood approximations based on prediction samples
3.6.3 Monte Carlo approximations based on smoother samples
3.6.4 Examples
3.6.5 Asymptotics of the maximum likelihood estimator
3.7 Extensions of the model
3.7.1 Spatial models
3.7.2 Stochastic context-free grammars
Acknowledgments
3.8 References
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CHAPTER 4: Monte Carlo Methods on Genetic Structures
4.1 Genetics, pedigrees, and structured systems
4. l. l Introduction t o Genetics
4.1.2 The conditional independence structures of genetics
4.1.3 โPeelingโ: sequential computation
4.1.4 The Baum algorithm for conditional probabilities
4.2 Computations on pedigrees
4.2.1 Peeling meiosis indicators
4.2.2 Peeling genotypes on pedigrees
4.2.3 Importance sampling and Monte Carlo likelihood
4.2.4 Risk, Elods, conditional Elods, and sequential imputation
Risk probabilities
Elods
Elods conditional on trait data
Monte Carlo likelihood by sequential imputation
4.2.5 Monte Carlo likelihood ratio estimation
Monte Carlo likelihood ratios
Monte Carlo likelihood surfaces
4.3 MCMC methods for Multilocus Genetic Data
4.3.1 Monte Carlo estimation of location score curves
4.3.2 Markov chain Monte Carlo
4.3.3 Single-site updating methods
4.3.4 Block updating; combining exact and MC computation
4.3.5 Tightly-linked loci: the M-sampler
4.4 Conclusion
Acknowledgement
4.5 References
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CHAPTER 5: Renormalization of Interacting Diffusions
5.1 Introduction
5.2 The model
5.3 Interpretation of the model
5.4 Block averages and renormalization
5.5 The hierarchical lattice
5.6 The renormalization transformation
5.7 Analysis of the orbit
5.8 Higher-dimensional state spaces
5.9 Open problems
5.10 Conclusion
5.11 References
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CHAPTER 6: Steinโs Method for Epidemic Processes
6.1 Introduction
6.2 A brief introduction to Steinโs method
6.2.1 Steinโs method for normal approximations
6.2.2 Steinโs method in general
6.2.3 Steinโs method for the weak law of large numbers
6.2.4 Steinโs method for the weak law in measure space
6.3 A bound on the distance of the GSE to its mean field limit
6.3.1 Assumptions
6.3.2 Heuristics
6.3.3 Previous results
Example: A Markovian epidemic
6.3.4 A bound on the distance to the: mean field limit
6.3.5 A special case: X(t,x) = ax(t)
6.3.6 Some plots of the limiting expression
6.4 Discussion
Acknowledgments
6.5 References