Bayesian Time Series Models

434 Pages·2011·9.08 MB·English

by  Barber D., Cemgil A.T., Chiappa S. (eds.)

BAYESIAN TIME SERIES MODELS
‘What’sgoingtohappennext?’Timeseriesdataholdtheanswers,andBayesianmethods
represent the cutting edge in learning what they have to say. This ambitious book is the
firstunifiedtreatmentoftheemergingknowledge-baseinBayesiantimeseriestechniques.
Exploiting the unifying framework of probabilistic graphical models, the book covers
approximation schemes, both Monte Carlo and deterministic, and introduces switching,
multi-object, nonparametric and agent-based models in a variety of application environ-
ments. It demonstrates that the basic framework supports the rapid creation of models
tailored to specific applications and gives insight into the computational complexity of
theirimplementation.
Theauthorsspantraditionaldisciplinessuchasstatisticsandengineeringandthemore
recentlyestablishedareasofmachinelearningandpatternrecognition.Readerswithabasic
understandingofappliedprobability,butnoexperiencewithtimeseriesanalysis,areguided
fromfundamentalconceptstothestateoftheartinresearchandpractice.

BAYESIAN TIME SERIES MODELS
Editedby
David Barber
UniversityCollegeLondon
A. Taylan Cemgil
Bog˘azic¸iUniversity,Bebek,Istanbul
Silvia Chiappa
UniversityofCambridge
cambridge university press
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CambridgeUniversityPress
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Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Bayesiantimeseriesmodels/editedbyDavidBarber,A.TaylanCemgil,SilviaChiappa.
p. cm.
ISBN978-0-521-19676-5(hardback)
1. Time-seriesanalysis. 2. Bayesianstatisticaldecisiontheory. I. Barber,David,1968Nov.9–
II. Cemgil,AliTaylan. III. Chiappa,Silvia. IV. Title.
QA280.B39 2011
519.5(cid:48)5–dc22
2011008051
ISBN978-0-521-19676-5Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
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Contents
Listofcontributors pagex
Preface xi
1 Inference and estimation in probabilistic time series models. David
Barber,A.TaylanCemgilandSilviaChiappa. 1
1.1 Timeseries 1
1.2 Markovmodels 3
1.3 LatentMarkovmodels 6
1.4 InferenceinlatentMarkovmodels 10
1.5 Deterministicapproximateinference 15
1.6 MonteCarloinference 21
1.7 Discussionandsummary 29
I MonteCarlo
2 Adaptive Markov chain Monte Carlo: theory and methods. Yves
Atchade´,GersendeFort,EricMoulinesandPierrePriouret. 32
2.1 Introduction 32
2.2 AdaptiveMCMCalgorithms 34
2.3 Convergenceofthemarginaldistribution 43
2.4 Stronglawoflargenumbers 44
2.5 Convergenceoftheequi-energysampler 46
2.6 Conclusion 47
2.A Appendix:ProofofSection2.5 47
3 Auxiliary particle filtering: recent developments. Nick Whiteley and
AdamM.Johansen. 52
3.1 Background 52
3.2 Interpretationandimplementation 56
3.3 Applicationsandextensions 63
3.4 FurtherstratifyingtheAPF 69
3.5 Conclusions 79
4 MonteCarloprobabilisticinferencefordiffusionprocesses:amethod-
ologicalframework.OmirosPapaspiliopoulos. 82
4.1 Introduction 82
4.2 Randomweightcontinuous–discreteparticlefiltering 84
4.3 Transitiondensityrepresentationforaclassofdiffusions 87
vi Contents
4.4 Exactsimulationofdiffusions 88
4.5 ExactsimulationofkilledBrownianmotion 90
4.6 Unbiasedestimationofthetransitiondensityusingseriesexpansions 92
4.7 Discussionanddirections 99
II Deterministicapproximations
5 Twoproblemswithvariationalexpectationmaximisationfortimeseries
models.RichardEricTurnerandManeeshSahani. 104
5.1 Introduction 104
5.2 Thevariationalapproach 104
5.3 Compactnessofvariationalapproximations 109
5.4 Variationalapproximationsarebiased 112
5.5 Conclusion 123
6 Approximate inference for continuous-time Markov processes. Ce´dric
ArchambeauandManfredOpper. 125
6.1 Introduction 125
6.2 Partlyobserveddiffusionprocesses 126
6.3 HiddenMarkovcharacterisation 127
6.4 Thevariationalapproximation 129
6.5 TheGaussianvariationalapproximation 133
6.6 Diffusionswithmultiplicativenoise 137
6.7 Parameterinference 137
6.8 Discussionandoutlook 138
7 Expectation propagation and generalised EP methods for inference in
switchinglineardynamicalsystems.OnnoZoeterandTomHeskes. 141
7.1 Introduction 141
7.2 Notationandproblemdescription 142
7.3 Assumeddensityfiltering 143
7.4 Expectationpropagation 145
7.5 Free-energyminimisation 148
7.6 Generalisedexpectationpropagation 150
7.7 Alternativebackwardpasses 153
7.8 Experiments 155
7.9 Discussion 159
7.A Appendix:OperationsonconditionalGaussianpotentials 160
7.B Appendix:ProofofTheorem7.1 163
8 Approximate inference in switching linear dynamical systems using
Gaussianmixtures.DavidBarber. 166
8.1 Introduction 166
8.2 Theswitchinglineardynamicalsystem 166
8.3 Gaussiansumfiltering 168
8.4 Expectationcorrection 172
8.5 Demonstration:trafficflow 176
8.6 Comparisonofsmoothingtechniques 177
8.7 Summary 180
Contents vii
III Switchingmodels
9 Physiological monitoring with factorial switching linear dynamical
systems.JohnA.QuinnandChristopherK.I.Williams. 182
9.1 Introduction 182
9.2 Model 184
9.3 Novelconditions 186
9.4 Parameterestimation 188
9.5 Inference 194
9.6 Experiments 197
9.7 Summary 203
10 Analysis of changepoint models. Idris A. Eckley, Paul Fearnhead and
RebeccaKillick. 205
10.1 Introduction 205
10.2 Singlechangepointmodels 207
10.3 Multiplechangepointmodels 210
10.4 Comparisonofmethods 215
10.5 Conclusion 221
10.A Appendix:segmentparameterestimation 222
IV Multi-objectmodels
11 Approximatelikelihoodestimationofstaticparametersinmulti-target
models.SumeetpalS.Singh,NickWhiteleyandSimonJ.Godsill. 225
11.1 Introduction 225
11.2 Themulti-targetmodel 228
11.3 AreviewofthePHDfilter 228
11.4 Approximatingthemarginallikelihood 232
11.5 SMCapproximationofthePHDfilteranditsgradient 233
11.6 Parameterestimation 238
11.7 Simulationstudy 239
11.8 Conclusion 243
12 Sequential inference for dynamically evolving groups of objects. Sze
KimPang,SimonJ.Godsill,JackLi,Franc¸oisSeptierandSimonHill. 245
12.1 Introduction 245
12.2 MCMC-particlesalgorithm 246
12.3 Grouptracking 258
12.4 Groundtargettracking 259
12.5 Groupstockselection 268
12.6 Conclusions 273
12.A Appendix:Basegrouprepresentation 275
13 Non-commutativeharmonicanalysisinmulti-objecttracking.RisiKon-
dor. 277
13.1 Introduction 277
13.2 Harmonicanalysisonfinitegroups 278
13.3 Band-limitedapproximations 282
viii Contents
13.4 AhiddenMarkovmodelinFourierspace 282
13.5 Approximationsintermsofmarginals 288
13.6 Efficientcomputation 290
13.7 Conclusions 293
V Nonparametricmodels
14 Markov chain Monte Carlo algorithms for Gaussian processes.
MichalisK.Titsias,MagnusRattrayandNeilD.Lawrence. 295
14.1 Introduction 295
14.2 Gaussianprocessmodels 297
14.3 Non-Gaussianlikelihoodsanddeterministicmethods 299
14.4 SamplingalgorithmsforGaussianprocessmodels 300
14.5 Relatedworkandothersamplingschemes 307
14.6 Demonstrationonregressionandclassification 308
14.7 Transcriptionalregulation 309
14.8 Dealingwithlargedatasets 312
14.9 Discussion 314
15 Nonparametric hidden Markov models. Jurgen Van Gael and Zoubin
Ghahramani. 317
15.1 Introduction 317
15.2 FromHMMstoBayesianHMMs 319
15.3 TheinfinitehiddenMarkovmodel 321
15.4 Inference 327
15.5 Example:unsupervisedpart-of-speechtagging 332
15.6 BeyondtheiHMM 334
15.7 Conclusions 337
15.A Appendix: Equivalence of the hierarchical Polya urn and hierarchical
Dirichletprocess 338
16 BayesianGaussianprocessmodelsformulti-sensortimeseriespredic-
tion. MichaelA.Osborne,AlexRogers,StephenJ.Roberts,SarvapaliD.
RamchurnandNickR.Jennings. 341
16.1 Introduction 341
16.2 Theinformationprocessingproblem 342
16.3 Gaussianprocesses 343
16.4 Trialimplementation 350
16.5 Empiricalevaluation 351
16.6 Computationtime 357
16.7 Relatedwork 358
16.8 Conclusions 359
16.A Appendix 360
VI Agent-basedmodels
17 Optimal control theory and the linear Bellman equation. Hilbert J.
Kappen. 363
17.1 Introduction 363