Highly Structured Stochastic Systems

, by Green, Peter J.; Hjort, Nils Lid; Richardson, Sylvia

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ISBN: 9780198510550 | 0198510551
Cover: Hardcover
Copyright: 7/3/2003

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Highly Structured Stochastic Systems (HSSS) is a modern strategy for building statistical models for challenging real-world problems, for computing with them, and for interpreting the resulting inference. The aim of this book is to make recent developments in HSSS accessible to a general statistical audience including graduate students and researchers.

List of contributors

Introducing Highly Structured Stochastic Systems

(1)

Peter J. Green

Nils Lid Hjort

Sylvia Richardson

How to read this book

(1)

What are highly structured stochastic systems?

(5)

Background

(1)

Hierarchical models

(1)

Conditional independence

(1)

Graphical models

(1)

The Bayesian paradigm

(1)

Computational methods

(1)

Cross-fertilization

(1)

The ESF-funded HSSS Programme

(2)

Structure of the book

(3)

Expectations of the future

(2)

Acknowledgements

(1)

References

(1)

Some modern applications of graphical models

(20)

Steffen L. Lauritzen

Introduction

(1)

Bayesian networks and probability propagation

(2)

Decision support and expert systems

(6)

Medical diagnosis

(2)

Troubleshooting with Bayesian networks

(1)

Decision networks

(3)

Digital communication

(4)

Coding and decoding noisy communication channels

(1)

Convolutional codes

(1)

Iterative decoding and turbocodes

(3)

Machine learning and variational inference

(2)

Other applications

(6)

References

(5)

Analysing social science data with graphical Markov models

(6)

Nanny Wermuth

Introduction

(1)

A motivating research example

(2)

Some consequences of directed acyclic graphs

(3)

Consequences in simple cases

(1)

Some consequences of large graphs

(2)

Relations to traditional methods and case studies

(1)

Acknowledgements

(1)

Analysis of DNA mixtures using Bayesian networks

(6)

Julia Mortera

Acknowledgement

(1)

Additional references in discussion

(2)

Causal inference using influence diagrams: the problem of partial compliance

(21)

A. Philip Dawid

Introduction

(1)

Causal models

(2)

Extended influence diagrams

(2)

Conditional independence

(1)

Partial compliance

(5)

Encouragement trial

(3)

Minimal representation

(1)

Constraints on φ

(2)

Causal inference

(6)

Intervention experiment

(1)

Combination

(2)

Inference

(1)

Bounding the intervention response probabilities

(2)

Bounds for α

(1)

Functional representation

(3)

Conclusions

(3)

Acknowledgements

(1)

References

(3)

Commentary: causality and statistics

(4)

Elja Arjas

Semantics of causal DAG models and the identification of direct and indirect effects

(13)

James M. Robins

Causal models and their DAG representation

(3)

Direct and indirect effects

(5)

Substantive considerations

(5)

Additional references in discussion

(3)

Causal inference via ancestral graph models

(23)

Thomas S. Richardson

Peter Spirtes

Introduction

(3)

Graphical models

(1)

Directed acyclic graphs and data-generating processes

(1)

Causal models

(1)

Inferring causal structure

(1)

The problem of unobserved variables

(1)

Ancestral graphs

(1)

Overview

(1)

Basic graphical terms and concepts

(1)

Graphical Markov models

(2)

Example: undirected graphical Markov models

(1)

Example: directed acyclic graphical Markov models

(1)

Ancestral graph Markov models

(4)

Definition of an ancestral graph

(1)

The global Markov property for ancestral graphs

(1)

Pairwise Markov property: maximal ancestral graphs

(1)

Gaussian parameterization via recursive equations with correlated errors

(1)

Gaussian independence models

(1)

Future work: the discrete case

(1)

Causal interpretation of DAG models

(2)

Causal DAGs

(1)

Examples

(1)

Identification of causal effects when the causal DAG is unknown

(8)

Marginalizing and conditioning independence models (J[SL)

(1)

Marginalizing and conditioning for ancestral graphs

(2)

Relating G and G[SL

(2)

Markov equivalence classes of maximal ancestral graphs

(1)

PAGs over four variables

100

(2)

Relation to other work

102

(1)

Discussion

102

(4)

Acknowledgements

103

(1)

References

103

(3)

Other approaches to the description of conditional independence structures

106

(3)

Milan Studeny

Introduction

106

(1)

Chain graphs

107

(1)

Annotated graphs

108

(1)

Structural imsets

108

(1)

Acknowledgements

108

(1)

On ancestral graph Markov models

109

(6)

Jan T. A. Koster

Additional references in discussion

112

(3)

Causality and graphical models in time series analysis

115

(23)

Rainer Dahlhaus

Michael Eichler

Introduction

115

(1)

Graphical models for multivariate time series

116

(5)

Time series chain graphs

117

(1)

Granger causality graphs

118

(2)

Partial correlation graphs

120

(1)

Markov properties

121

(5)

Global Markov properties for PC graphs

122

(1)

Global Markov properties for TSC graphs

122

(2)

Global Markov properties for causality graphs

124

(2)

The p-separation criterion

126

(1)

Inference for graphical time series models

126

(3)

Conditional likelihood for graphical autoregressive models

126

(1)

Partial directed and partial contemporaneous correlation

127

(1)

Partial correlation graphs

128

(1)

Applications

129

(4)

Application to air pollution data

129

(2)

Application to human tremor data

131

(2)

Discussion

133

(5)

References

134

(4)

Graphical models for stochastic processes

138

(3)

Vanessa Didelez

Introduction

138

(1)

Granger causality versus intervention causality

138

(1)

The continuous-time situation

139

(2)

Conditional association in time series

141

(4)

Hans R. Kunsch

Concepts of conditional association in time series

141

(1)

Discovery and interpretation of structure

142

(1)

Reducing the number of parameters

143

(1)

Model fitting

143

(2)

Additional references in discussion

144

(1)

Linking theory and practice of MCMC

145

(22)

Gareth O. Roberts

Introduction

145

(2)

Motivating example

147

(3)

Mathematical framework

150

(2)

Rates of convergence of Markov chains

152

(1)

Drift and minorization conditions

152

(3)

Central limit theorems

154

(1)

MCMC and drift conditions

155

(4)

Practical implications

156

(2)

Other algorithms

158

(1)

Examples

159

(1)

Computable bounds

159

(3)

Conclusions

162

(5)

Acknowledgements

164

(1)

References

164

(3)

Advances in MCMC: a discussion

167

(5)

Christian P. Robert

Introduction

167

(1)

Parameterization and reparameterization

167

(3)

Perfect sampling

170

(1)

Conclusion

171

(1)

Acknowledgements

171

(1)

On some current research in MCMC

172

(7)

Arnoldo Frigessi

Introduction

172

(1)

Adaptive MCMC

172

(2)

Non-reversible Markov chains

174

(1)

Post-processing

175

(1)

More ideas

175

(4)

Acknowledgements

175

(1)

Additional references in discussion

175

(4)

Trans-dimensional Markov chain Monte Carlo

179

(20)

Peter J. Green

Introduction

179

(1)

Reversible jump MCMC

180

(4)

Metropolis-Hastings on a general state space

181

(1)

A constructive representation in terms of random numbers

182

(1)

The trans-dimensional case

182

(1)

Details of application to the model-choice problem

183

(1)

Some remarks and ramifications

183

(1)

Relations to other across-model approaches

184

(2)

Jump diffusion

184

(1)

Point processes, with and without marks

185

(1)

Product-space formulations

185

(1)

Alternatives to joint model-parameter sampling

186

(2)

Integrating out the parameters

186

(1)

Within-model simulation

187

(1)

Some issues in choosing a sampling strategy

188

(3)

Is it good to jump?

189

(1)

The two-model case

189

(1)

Finite state space example

190

(1)

Tempering-by-embedding

190

(1)

An automatic generic trans-dimensional sampler

191

(3)

Examples

193

(1)

Limitations of this approach

194

(1)

Methodological extensions

194

(5)

Delayed rejection

194

(1)

Efficient proposal choice for reversible jump MCMC

195

(1)

Diagnostics for reversible jump MCMC

195

(1)

Acknowledgements

196

(1)

References

196

(3)

Proposal densities and product-space methods

199

(4)

Simon J. Godsill

Introduction

199

(1)

Construction of proposal densities

199

(2)

Product-space methods

201

(2)

Trans-dimensional Bayesian non-parametrics with spatial point processes

203

(4)

Juha Heikkinen

Introduction

203

(1)

Non-parametric Bayesian function estimation

203

(2)

On constraints and dimension matching

205

(2)

Acknowledgements

206

(1)

Additional references in discussion

206

(1)

Particle filtering methods for dynamic and static Bayesian problems

207

(21)

Carlo Berzuini

Walter R. Gilks

Introduction

207

(1)

Dynamic Bayesian problems

208

(1)

Motivating example

209

(2)

General particle filtering strategy

211

(2)

The Sampling Importance Resampling filter

213

(1)

Why resample?

214

(1)

Efficient resampling

215

(1)

Improverishment

215

(1)

Introducing Markov iterations

216

(1)

Enhancing particle diversity (resampling and moving)

217

(4)

An illustration

221

(1)

Particle filtering for static problems

222

(6)

Acknowledgements

224

(1)

References

224

(4)

Some further topics on Monte Carlo methods for dynamic Bayesian problems

228

(4)

Geir Storvik

Introduction

228

(1)

When to resample

228

(2)

Theoretical properties of Monte Carlo filters

230

(1)

Static parameters

231

(1)

General principles in sequential Monte Carlo methods

232

(5)

Peter Clifford

Particle filters

232

(2)

Generalizations and enhancements

234

(2)

Monte Carlo integration

234

(1)

Regularized sampling and effective dimension

234

(1)

Parametric particle filters

234

(1)

Self-avoiding particles

235

(1)

Bearings-only tracking

236

(1)

Additional references in discussion

236

(1)

Spatial models in epidemiological applications

237

(23)

Sylvia Richardson

Introduction

237

(2)

Level of spatial modelling for disease outcomes

239

(1)

Types of data

239

(1)

Hierarchical models

240

(1)

Models for point data

240

(4)

Framework

240

(1)

Inhomogeneous Poisson processes

241

(1)

Cox processes

242

(1)

Spatial variability of the disease rate

243

(1)

Aggregation of point level models

244

(2)

Stratification

244

(1)

Models linked to Gaussian random fields (GRFs)

245

(1)

Models linked to gamma random fields

245

(1)

Models directly specified at the aggregated level

246

(3)

Markov random fields models

246

(1)

Direct modelling of covariance structure

247

(1)

Partition, allocation, and mixture models

248

(1)

Covariate effects in spatial models

249

(5)

Point process models

250

(1)

Aggregation and ecological bias

251

(1)

Covariate effects in models specified at the aggregated level

252

(1)

Confounding between covariates and random effects

252

(2)

Concluding remarks

254

(6)

Acknowledgements

255

(1)

References

255

(5)

Some remarks on Gaussian Markov random field models for disease mapping

260

(4)

Leonhard Knorr-Held

Acknowledgements

264

(1)

A comparison of spatial point process models in epidemiological applications

264

(7)

Jesper Moller

Brief description of models

264

(1)

General properties of the models

265

(2)

Aggregation coherence and natural scaling

267

(1)

Computational aspects

267

(1)

Concluding remarks

268

(3)

Acknowledgements

268

(1)

Additional references in discussion

268

(3)

Spatial hierarchical Bayesian models in ecological applications

271

(18)

Antti Penttinen

Fabio Divino

Anne Riiali

Introduction

271

(2)

Modelling the suitability of sites

273

(2)

Extension of the suitability model to two species

275

(1)

A conditional model for association of species

276

(4)

Empirical results

280

(4)

Discussion

284

(5)

Acknowledgements

286

(1)

References

286

(3)

Likelihood analysis of binary data in space and time

289

(6)

Julian Besag

Jeremy Tantrum

Introduction

289

(1)

Markov random field formulation

290

(1)

Parameter estimation

291

(3)

Maximum pseudo-likelihood estimates for the θtS

291

(1)

Maximum likelihood estimates for the θts

292

(2)

Time homogeneity

294

(1)

Discussion

294

(1)

Acknowledgements

295

(1)

Some further aspects of spatio-temporal modelling

295

(6)

Alexandra Mello Schmidt

Introduction

295

(1)

Modelling of heterogeneous spatial processes

295

(2)

Non-separable covariance matrices

297

(4)

Acknowledgement

298

(1)

Additional references in discussion

298

(3)

Advances in Bayesian image analysis

301

(22)

Merrilee Hurn

Oddvar Husby

Havard Rue

Introduction

301

(1)

Modelling

301

(8)

Prior modelling

302

(7)

Data modelling

309

(1)

Treatment of parameters

309

(2)

Estimation techniques for low-level models

310

(1)

Estimation techniques for high-level models

310

(1)

Fully Bayesian approaches

310

(1)

Point estimation for images

311

(1)

Example

312

(11)

A model for diffuse scattering in ultrasound imaging

313

(1)

Image restoration and MCMC algorithms

314

(1)

Contour detection

315

(1)

Acknowledgements

316

(1)

References

316

(7)

Probabilistic image modelling

323

(3)

M. N. M. van Lieshout

Introduction

323

(1)

Pixel-level modelling

323

(1)

High-level modelling

324

(1)

Concluding remarks

325

(1)

Prospects in Bayesian image analysis

326

(7)

Alain Trubuil

Introduction

326

(1)

Texture modelling

326

(1)

Level curves and segmentation

327

(1)

Applications

328

(5)

Additional references in discussion

330

(3)

Preventing epidemics in heterogeneous environments

333

(21)

Niels G. Becker

Sergey Utev

Introduction

333

(2)

The standard epidemic model

335

(3)

Model formulation

335

(1)

A threshold parameter

336

(1)

Critical immunity coverage

336

(1)

Estimation

337

(1)

Dependence, heterogeneity, and structure

338

(1)

Variation in infectivity and susceptibility

338

(8)

Unidentifiable types

338

(3)

Identifiable types

341

(2)

Variable vaccine response

343

(3)

Community of households

346

(2)

Contact networks and other models

348

(6)

Disease transmission via contact sets

348

(1)

Variation over time

349

(1)

Other models

350

(1)

Acknowledgements

350

(1)

References

350

(4)

MCMC methods for stochastic epidemic models

354

(4)

Philip D. O'Neil

Introduction

354

(1)

Advantages and challenges of MCMC

354

(1)

Examples

355

(3)

Temporal data, general stochastic epidemic

355

(1)

Temporal data, network model

356

(1)

Final size data, households model

356

(1)

Future directions

357

(1)

Towards Bayesian inference in epidemic models

358

(5)

Kari Auranen

Additional references in discussion

361

(2)

Genetic linkage analysis using Markov chain Monte Carlo techniques

363

(19)

Simon C. Heath

Introduction

363

(4)

Markers and trait loci

365

(1)

The computational difficulties of linkage analysis

365

(1)

Relationship of linkage to linkage disequilibrium analysis

366

(1)

Exact probability calculations

367

(6)

The peeling algorithm

367

(2)

The Lander-Green algorithm

369

(4)

Markov chain Monte Carlo sampling schemes

373

(4)

Single locus genotype samplers

373

(1)

Multilocus genotype samplers

374

(1)

Sampling segregation indicators

375

(1)

General genetic samplers

375

(1)

Samplers for imperfect data

376

(1)

Sampling trait locus number

377

(1)

Discussion

377

(5)

References

378

(4)

Graphical models for mapping continuous traits

382

(4)

Nuala A. Sheehan

Daniel A. Sorensen

Introduction

382

(1)

QTL mapping on a granddaughter design

382

(4)

Conclusion

386

(1)

Acknowledgements

386

(1)

Statistical approaches to genetic mapping

386

(7)

David A. Stephens

Association mapping and genome-wide maps

388

(1)

Future developments: data mining approaches to genetic mapping

389

(4)

Additional references in discussion

391

(2)

The genealogy of a neutral mutation

393

(21)

R. C. Griffiths

Simon Tavare

Introduction

393

(3)

Coalescent trees

394

(1)

Variable population size

395

(1)

Relationship between coalescent trees and urn models

396

(3)

The number of descendants of an edge

397

(1)

Coalescence times in a subtree

397

(1)

Branches in the subtree below a mutation

398

(1)

Mutations in the tree

399

(2)

The distribution of the number of segregating sites

400

(1)

The distribution of pairwise differences

400

(1)

Tajima's D

400

(1)

Frequency spectra

401

(1)

Distribution of the age of a mutation

402

(1)

Coalescence times in a subtree

403

(2)

Further mutations in subtrees

405

(2)

A simulation algorithm

405

(1)

The age of the mutation

405

(1)

An example of simulation of ages

406

(1)

Simulation of Tajima's D in a subtree

407

(1)

Sampling under a mutation in the population

407

(2)

Results for the standard coalescent

408

(1)

Other sampling schemes

409

(2)

Sampling from the disease population

409

(1)

Case-control sampling

410

(1)

Conclusion

411

(3)

Acknowledgements

411

(1)

References

412

(2)

Linked versus unlinked DNA data - a comparison based on ancestral inference

414

(3)

Gunter Weiss

Estimation of a scaled mutation rate parameter

414

(1)

Estimation of a growth rate parameter

415

(1)

An illustrative example

416

(1)

The age of a rare mutation

417

(6)

Carsten Wiuf

Introduction

417

(2)

Defining the problem

419

(1)

The likelihood approach

419

(1)

The coalescent approach

420

(1)

Conclusion

421

(2)

Additional references in discussion

421

(2)

HSSS model criticism

423

(22)

Anthony O'Hagan

Introduction

423

(5)

The role of model criticism

423

(1)

Model adequacy and sensitivity analysis

424

(1)

Model criticism

425

(2)

The origins and structure of this article

427

(1)

Predictive diagnostic techniques

428

(3)

Comparing data with model predictions

428

(2)

Predictive diagnostics

430

(1)

Node-based methods

431

(9)

Residuals

432

(2)

Conflict

434

(2)

Example

436

(2)

Interpretation of conflict

438

(1)

Some related issues

439

(1)

Sensitivity to prior distributions

440

(3)

Diagnosing potential sensitivity

440

(2)

Example

442

(1)

Discussion

443

(2)

Acknowledgements

443

(1)

References

443

(2)

Which `base' distribution for model criticism?

445

(4)

M. J. Bayarri

Basic issues in model criticism

445

(1)

Distributions for the diagnostic measure

446

(1)

An example (Bayarri and Morales 2003)

447

(1)

Concluding remarks

448

(1)

Some comments on model criticism

449

(6)

Alan E. Gelfand

Additional references in discussion

452

(3)

Topics in non-parametric Bayesian statistics

455

(24)

Nils Lid Hjort

Introduction and summary

455

(2)

What is it, and why?

455

(2)

On the present article

457

(1)

The Dirichlet process prior and some extensions

457

(2)

The Dirichlet process

457

(1)

The Dirichlet as a limit

458

(1)

An extension via an infinite sum representation

459

(1)

Random Dirichlet means

459

(2)

Transform identities

459

(1)

Stochastic equations and the full moment sequence

460

(1)

Quantile pyramid processes

461

(3)

Quantile inference with the Dirichlet process

461

(1)

An automatic non-parametric density estimator

462

(1)

Quantile pyramids

463

(1)

Posterior quantile pyramids

463

(1)

Bayesian density estimation and consistency issues

464

(2)

Levy frailty processes and proportional hazards

466

(2)

Beta processes in a linear hazard model

468

(1)

Local Bayesian regression

469

(2)

Random shapes with smoothed Dirichlets

471

(1)

Non-parametric envelopes around parametric models

472

(2)

A semiparametric Bayes model

472

(1)

Model fitting with control sets

473

(1)

Randomness around a parametric survival data model

473

(1)

Concluding remarks

474

(5)

Acknowledgements

474

(1)

References

475

(4)

Asymptotics of non-parametric posteriors

479

(4)

Aad van der Vaart

Introduction

479

(1)

Questions

479

(2)

Semiparametric versus non-parametric

479

(1)

Really high-dimensional problems

480

(1)

Proper and fixed

481

(1)

Model selection

481

(1)

Rates of convergence

481

(2)

A predictive point of view on Bayesian non-parametrics

483

(6)

Sonia Petrone

An example

483

(1)

A predictive approach

484

(1)

Non-parametric priors for high-dimensional data

484

(1)

Characterization of the prior via predictive assumptions

484

(1)

Goodness of fit

485

(1)

Computational issues and asymptotic behaviour

485

(1)

Problems of dependence

486

(3)

Additional references in discussion

486

(3)

Author index

489

(14)

Subject index

503

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Highly Structured Stochastic Systems

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