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E-raamat: Handbook of Graphical Models

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A graphical model is a statistical model that is represented by a graph. The factorization properties underlying graphical models facilitate tractable computation with multivariate distributions, making the models a valuable tool with a plethora of applications. Furthermore, directed graphical models allow intuitive causal interpretations and have become a cornerstone for causal inference.

While there exist a number of excellent books on graphical models, the field has grown so much that individual authors can hardly cover its entire scope. Moreover, the field is interdisciplinary by nature. Through chapters by leading researchers from different areas, this handbook provides a broad and accessible overview of the state of the art.

Key features:

* Contributions by leading researchers from a range of disciplines

* Structured in five parts, covering foundations, computational aspects, statistical inference, causal inference, and applications

* Balanced coverage of concepts, theory, methods, examples, and applications

* Chapters can be read mostly independently, while cross-references highlight connections

The handbook is targeted at a wide audience, including graduate students, applied researchers, and experts in graphical models.

Arvustused

"The Handbook of Graphical Models is an edited collection of chapters written by leading researchers and covering a wide range of topics on probabilistic graphical models. The editors, Marloes Maathuis, Mathias Drton, Steffen Lauritzen, and Martin Wainwright, are well-known statisticians and have conducted foundational research on graphical models. They have done a great job of soliciting and organizing chapters authored by top researchers from a variety of disciplines beyond just mathematics, probability and statistics; many authors hail from computer science, electrical engineering, economics, and even philosophy. It is precisely the multidisciplinary nature of this book that makes it stand out from other texts on graphical models. Because of this, the Handbook of Graphical Models will have broad appeal across many disciplines, providing a unique resource and excellent reference for those researching, studying, and using graphical models...Overall, the Handbook of Graphical Models is an important reference on probabilistic graphical models that will be used by researchers in statistics and probability, computer science, electrical engineering and beyond. The book stands out for its broad, multidisciplinary nature, with wide-ranging and largely theoretical coverage of core topics and the latest research on graphical models." - Genevera I. Allen, JASA, August 2020

Preface xv
Contributors xvii
I Conditional independencies and Markov properties 1(80)
1 Conditional Independence and Basic Markov Properties
3(36)
Milan Studeny
1.1 Introduction: Historical Overview and an Example
4(4)
1.1.1 Stochastic conditional independence
4(1)
1.1.2 Graphs and local computation method
4(1)
1.1.3 Conditional independence in other areas
5(1)
1.1.4 Geometric approach and methods of modern algebra
5(1)
1.1.5 A motivational example
6(2)
1.2 Notation and Elementary Concepts
8(3)
1.2.1 Discrete probability measures
8(2)
1.2.2 Continuous distributions
10(1)
1.3 The Concept of Conditional Independence
11(4)
1.3.1 Conditional independence in the discrete case
11(3)
1.3.2 More general CI concepts
14(1)
1.4 Basic Properties of Conditional Independence
15(3)
1.4.1 Conditional independence structure
15(2)
1.4.2 Statistical model of a CI structure
17(1)
1.5 Semi-graphoids, Graphoids, and Separoids
18(3)
1.5.1 Elementary and dominant triplets
19(2)
1.6 Elementary Graphical Concepts
21(1)
1.7 Markov Properties for Undirected Graphs
22(2)
1.7.1 Global Markov property for an UG
22(1)
1.7.2 Local and pairwise Markov properties for an UG
23(1)
1.7.3 Factorization property for an UG
24(1)
1.8 Markov Properties for Directed Graphs
24(7)
1.8.1 Directional separation criteria
24(5)
1.8.2 Global Markov property for a DAG
29(1)
1.8.3 Local Markov property for a DAG
29(1)
1.8.4 Factorization property for a DAG
30(1)
1.8.5 Markov equivalence for DAGs
30(1)
1.9 Remarks on Chordal Graphs
31(1)
1.10 Imsets and Geometric Views
32(1)
1.10.1 The concept of a structural imset
32(1)
1.10.2 Imsets for statistical learning
33(1)
1.11 CI Inference
33(6)
2 Markov Properties for Mixed Graphical Models
39(22)
Robin Evans
2.1 Introduction
39(4)
2.1.1 Decomposable graphs
40(1)
2.1.2 Unification
41(1)
2.1.3 Marginalizing and conditioning
41(1)
2.1.4 Outline of the chapter
42(1)
2.2 Chain Graphs
43(4)
2.2.1 Factorization
44(1)
2.2.2 Local Markov property
45(1)
2.2.3 General remarks
46(1)
2.2.4 Other Markov properties for chain graphs
46(1)
2.3 Closed Independence Models
47(4)
2.3.1 ADMGs
47(1)
2.3.2 Ancestral sets
48(1)
2.3.3 Districts
49(1)
2.3.4 A conditional independence model
50(1)
2.3.5 Connection to chain graphs
51(1)
2.4 Non-Independence Constraints
51(4)
2.4.1 Verma constraints
52(2)
2.4.2 Inequalities
54(1)
2.4.3 mDAGs
54(1)
2.5 Other Graphs and Models
55(3)
2.5.1 Other models
55(1)
2.5.2 Ancestral graphs
56(1)
2.5.3 Quantum states
57(1)
2.6 Summary
58(3)
3 Algebraic Aspects of Conditional Independence and Graphical Models
61(20)
Thomas Kahle
Johannes Rauh
Seth Sullivant
3.1 Introduction
61(2)
3.2 Notions of Algebraic Geometry and Commutative Algebra
63(5)
3.2.1 Polynomials, ideals and varieties
63(2)
3.2.2 Irreducible and primary decomposition
65(2)
3.2.3 Binomial ideals
67(1)
3.2.4 Real algebraic geometry
67(1)
3.3 Conditional Independence Ideals
68(5)
3.3.1 Discrete random variables
68(3)
3.3.2 Gaussian random variables
71(1)
3.3.3 The contraction axiom
72(1)
3.4 Examples of Decompositions of Conditional Independence Ideals
73(2)
3.4.1 The intersection axiom
73(1)
3.4.2 The four-cycle
74(1)
3.5 The Vanishing Ideal of a Graphical Model
75(3)
3.6 Further Reading
78(3)
II Computing with factorizing distributions 81(108)
4 Algorithms and Data Structures for Exact Computation of Marginals
83(34)
Jeffrey A. Bilmes
4.1 Introduction
84(2)
4.1.1 Graphical models
84(1)
4.1.2 Probabilistic inference
85(1)
4.1.3 Benefits of graphical models for inference
86(1)
4.2 Inference on Trees
86(5)
4.2.1 Trees and tree structured factorization
87(1)
4.2.2 Eliminating variables via marginalization
88(2)
4.2.3 The variable elimination process on graphs
90(1)
4.3 Triangulated Graphs and Fill-in Free Elimination Orders
91(10)
4.3.1 Graphs with fill-in free orders
92(1)
4.3.2 Triangulated graphs
92(3)
4.3.3 Good heuristics for choosing an elimination order
95(1)
4.3.4 The running intersection property and junction trees
96(4)
4.3.5 Entanglement
100(1)
4.4 Inference on Junction Trees
101(10)
4.4.1 Benefits of junction trees
101(1)
4.4.2 Factorization
102(1)
4.4.3 Potentials as true marginals
103(1)
4.4.4 Message initialization
103(2)
4.4.5 Necessary condition for true marginals
105(1)
4.4.6 Achieving true marginals via message passing
106(4)
4.4.7 Message schedules
110(1)
4.5 Discussion
111(6)
5 Approximate Methods for Calculating Marginals and Likelihoods
117(24)
Nicholas Ruozzi
5.1 Inference as Optimization
118(2)
5.1.1 The Kullback-Leibler divergence
118(1)
5.1.2 The Gibbs free energy
119(1)
5.2 The Bethe Free Energy
120(5)
5.2.1 Convex and reweighted free energies
122(1)
5.2.2 A combinatorial characterization of the Bethe free energy
123(2)
5.3 Algorithms for Approximate Marginal Inference
125(4)
5.3.1 Loopy belief propagation
125(1)
5.3.2 Reweighted message-passing algorithms
126(1)
5.3.3 Gradient ascent
127(1)
5.3.4 Naive mean field
127(1)
5.3.5 Sampling methods
128(1)
5.4 Approximate Learning
129(7)
5.4.1 Log-linear models
129(1)
5.4.2 Maximum likelihood estimation (MLE)
130(4)
5.4.3 Maximum entropy
134(1)
5.4.4 Pseudolikelihood learning
135(1)
5.5 Conclusion
136(1)
Appendix: Marginal Reparameterization of a Tree-Structured Distribution
136(5)
6 MAP Estimation: Linear Programming Relaxation and Message-Passing Algorithms
141(24)
Ofer Meshi
Alexander G. Schwing
6.1 Introduction
141(1)
6.2 The MAP Estimation Problem
142(2)
6.3 Sampling and Search-Based Methods
144(1)
6.3.1 Sampling
144(1)
6.3.2 Global search methods
144(1)
6.3.3 Local search methods
144(1)
6.4 Integer Programming and LP Relaxations
145(3)
6.4.1 LP relaxations
146(1)
6.4.2 Tight LP relaxations
147(1)
6.5 Optimization of the LP Relaxation
148(8)
6.5.1 The dual program
148(1)
6.5.2 Subgradient descent
148(1)
6.5.3 Block coordinate minimization
149(1)
6.5.4 e-descent
150(1)
6.5.5 The smoothed dual
151(2)
6.5.6 The strongly-convex dual
153(3)
6.5.7 Other optimization schemes
156(1)
6.6 Relation to Message-Passing Algorithms
156(1)
6.7 Rounding Schemes
157(1)
6.8 Conclusion
158(1)
Appendix: The Dual LP Relaxation
158(7)
7 Sequential Monte Carlo Methods
165(24)
Arnaud Doucet
Anthony Lee
7.1 Introduction
165(1)
7.2 Hidden Markov Models
166(3)
7.3 Particle Filtering and Smoothing
169(3)
7.4 Sequential Monte Carlo
172(4)
7.4.1 A general construction
172(1)
7.4.2 Convergence results
173(2)
7.4.3 Variance estimation
175(1)
7.5 Methodological Innovations
176(4)
7.5.1 Resampling schemes
176(1)
7.5.2 Auxiliary particle filters
177(1)
7.5.3 Reducing interaction for distributed implementation
178(1)
7.5.4 SMC samplers
179(1)
7.5.5 Alternative perspectives
180(1)
7.6 Particle MCMC
180(3)
7.7 Discussion
183(6)
III Statistical inference 189(162)
8 Discrete Graphical Models and Their Parameterization
191(26)
Luca La Rocca
Alberto Roverato
8.1 Introduction
191(1)
8.2 Notation and Terminology
192(2)
8.3 Overview of Discrete Graphical Models
194(3)
8.4 Basic Lemmas
197(3)
8.4.1 Establishing independence relationships
197(1)
8.4.2 Two properties of Mobius inversion
198(2)
8.5 Undirected Graph Models
200(3)
8.6 Bidirected Graph Models
203(3)
8.7 Regression Graph Models
206(4)
8.8 Non-binary Variables and Likelihood Inference
210(7)
9 Gaussian Graphical Models
217(22)
Caroline Uhler
9.1 The Gaussian Distribution and Conditional Independence
219(1)
9.2 The Gaussian Likelihood and Convex Optimization
220(3)
9.3 The MLE as a Positive Definite Completion Problem
223(1)
9.4 ML Estimation and Convex Geometry
224(3)
9.5 Existence of the MLE for Various Classes of Graphs
227(3)
9.6 Algorithms for Computing the MLE
230(3)
9.7 Learning the Underlying Graph
233(1)
9.8 Other Gaussian Models with Linear Constraints
234(5)
10 Bayesian Inference in Graphical Gaussian Models
239(26)
Helene Massam
10.1 Introduction
239(2)
10.2 Preliminaries
241(3)
10.2.1 Graphs and Markov properties
241(1)
10.2.2 The Wishart distribution
242(2)
10.3 Decomposable Graphs and the Hyper Inverse Wishart
244(7)
10.3.1 The graphical Gaussian (or concentration graph) model
244(1)
10.3.2 The hyper inverse Wishart prior
245(3)
10.3.3 Priors with several shape parameters
248(2)
10.3.4 Covariance graph models
250(1)
10.4 Arbitrary Undirected Graphs and the G-Wishart
251(5)
10.4.1 Computing the normalizing constant of the G-Wishart
251(1)
10.4.2 Sampling from the G-Wishart
252(1)
10.4.3 Moving away from Bayes factors
253(1)
10.4.4 Moving away from Bayes factors and the G-Wishart
254(2)
10.5 Matrix Variate Graphical Gaussian Models
256(2)
10.6 Fractional Bayes Factors
258(2)
10.7 Two Interesting Questions
260(5)
11 Latent Tree Models
265(24)
Piotr Zwiernik
11.1 Basics
266(5)
11.1.1 Definitions
266(1)
11.1.2 Motivation and applications
267(2)
11.1.3 Parsimonious latent tree models
269(1)
11.1.4 Gaussian and general Markov models
270(1)
11.2 Second-Order Moment Structure
271(5)
11.2.1 Gaussian latent tree model
271(2)
11.2.2 General Markov models
273(1)
11.2.3 Linear models
274(1)
11.2.4 Distance based methods
275(1)
11.3 Selected Theoretical Results
276(3)
11.3.1 Identifiability
276(1)
11.3.2 Guarantees for tree reconstruction
277(1)
11.3.3 Model selection
278(1)
11.4 Estimation and Inference
279(2)
11.4.1 Fixed tree structure
279(1)
11.4.2 The structural EM algorithm
279(1)
11.4.3 Phylogenetic invariants
280(1)
11.5 Discussion
281(8)
12 Neighborhood Selection Methods
289(20)
Po-Ling Loh
12.1 Introduction
289(1)
12.2 Notation
290(1)
12.3 Gaussian Graphical Models
290(6)
12.3.1 Inverse covariance matrix and edge structure
291(1)
12.3.2 Edge recovery via matrix estimation
291(1)
12.3.3 Edge recovery via linear regression
292(1)
12.3.4 Statistical theory
293(3)
12.4 Ising Models
296(3)
12.4.1 Logistic regression
297(1)
12.4.2 Other methods
298(1)
12.5 Generalizations and Extensions
299(3)
12.5.1 Nonparanormal distributions
299(1)
12.5.2 Augmented inverse covariance matrices
300(1)
12.5.3 Other exponential families
301(1)
12.6 Robustness
302(2)
12.6.1 Noisy and missing data
303(1)
12.6.2 Corrected graphical Lasso
303(1)
12.6.3 Latent variables
303(1)
12.7 Further Reading
304(5)
13 Nonparametric Graphical Models
309(16)
Han Liu
John Lafferty
13.1 Introduction
309(2)
13.2 Semiparametric Exponential Family Graphical Models
311(2)
13.2.1 Examples
312(1)
13.2.2 A nuisance-free loss function
313(1)
13.3 Tree and Forest Graphical Models
313(2)
13.3.1 Tree estimation
314(1)
13.3.2 Oracle properties
315(1)
13.4 Gaussian Copulas and Variants
315(4)
13.4.1 Estimation
317(1)
13.4.2 Rank correlation
318(1)
13.4.3 Trees and copulas
318(1)
13.5 Tensor Product Smoothing Spline ANOVA Models
319(3)
13.5.1 Tensor product smoothing splines
320(1)
13.5.2 Fisher-Hyvarinen scoring
320(2)
13.6 Summary and Extensions
322(3)
14 Inference in High-Dimensional Graphical Models
325(26)
Jana Jankova
Sara van de Geer
14.1 Undirected Graphical Models
325(13)
14.1.1 Introduction
325(2)
14.1.2 De-biasing regularized estimators
327(1)
14.1.3 Graphical Lasso
328(5)
14.1.4 Nodewise square-root Lasso
333(2)
14.1.5 Computational view
335(1)
14.1.6 Simulation results
335(2)
14.1.7 Discussion
337(1)
14.2 Directed Acyclic Graphs
338(3)
14.2.1 Maximum likelihood estimator with to-penalization
339(1)
14.2.2 Inference for edge weights
340(1)
14.3 Conclusion
341(1)
14.4 Proofs
342(11)
14.4.1 Proofs for undirected graphical models
342(4)
14.4.2 Proofs for directed acyclic graphs
346(5)
IV Causal inference 351(120)
15 Causal Concepts and Graphical Models
353(28)
Vanessa Didelez
15.1 Introduction
353(2)
15.2 Association versus Causation: Seeing versus Doing
355(3)
15.3 Extending Graphical Models for Causal Reasoning
358(6)
15.3.1 Intervention graphs
358(2)
15.3.2 Causal DAGs
360(3)
15.3.3 Comparison
363(1)
15.4 Graphical Rules for the Identification of Causal Effects
364(5)
15.4.1 The Back-Door Theorem
364(3)
15.4.2 The Front-Door Theorem
367(2)
15.5 Graphical Characterization of Sources of Bias
369(6)
15.5.1 Confounding
369(4)
15.5.2 Selection bias
373(2)
15.6 Discussion and Outlook
375(6)
16 Identification in Graphical Causal Models
381(24)
Ilya Shpitser
16.1 Introduction
381(1)
16.2 Causal Models of a DAG
382(6)
16.2.1 Causal, direct, indirect, and path-specific effects
384(1)
16.2.2 Responses to dynamic treatment regimes
385(1)
16.2.3 Identifiability
386(1)
16.2.4 Identification of causal effects
386(1)
16.2.5 Identification of path-specific effects
387(1)
16.2.6 Identification of responses to dynamic treatment regimes
388(1)
16.3 Causal Models of a DAG with Hidden Variables
388(8)
16.3.1 Latent projections and targets of inference
389(1)
16.3.2 Conditional mixed graphs and kernels
389(1)
16.3.3 The fixing operation
390(1)
16.3.4 The ID algorithm
391(3)
16.3.5 Controlled direct effects
394(1)
16.3.6 Conditional causal effects
394(1)
16.3.7 Path-specific effects
394(1)
16.3.8 Responses to dynamic treatment regimes
395(1)
16.4 Linear Structural Equation Models
396(4)
16.4.1 Global identification of linear SEMs
397(1)
16.4.2 Generic identification of linear SEMs
398(2)
16.5 Summary
400(5)
17 Mediation Analysis
405(34)
Johan Steen
Stijn Vansteelandt
17.1 Introduction
406(1)
17.2 Definitions and Notation
407(2)
17.2.1 Natural direct and indirect effects
407(1)
17.2.2 Path-specific effects
408(1)
17.3 Cross-World Quantities Call for Cross-World Assumptions
409(3)
17.3.1 Imposing cross-world independence
410(1)
17.3.2 Cross-world independence and NPSEMs
411(1)
17.3.3 Single world versus multiple worlds models
411(1)
17.3.4 Further outline
412(1)
17.4 Identification 1.0
412(7)
17.4.1 Unmeasured mediator-outcome confounding
412(1)
17.4.2 Adjusting for mediator-outcome confounding
413(1)
17.4.3 Treatment-induced mediator-outcome confounding
414(1)
17.4.4 Pearl's graphical criteria for conditional cross-world independence
415(1)
17.4.5 Sufficient conditions to recover natural effects from experimental data
415(1)
17.4.6 Sufficient conditions to recover natural effects from observational data
416(3)
17.5 Identification 2.0
419(7)
17.5.1 Building blocks for complete graphical identification criteria
420(3)
17.5.2 The central notion of recantation
423(3)
17.6 Complementary Identification Strategies
426(2)
17.6.1 Interchanging cross-world assumptions
426(1)
17.6.2 Two types of auxiliary variables
426(1)
17.6.3 Mediating instruments - some reasons for skepticism
427(1)
17.7 From Mediating Instruments to Conceptual Clarity
428(2)
17.7.1 In search of operational definitions
428(1)
17.7.2 Deterministic expanded graphs
428(1)
17.7.3 Some examples
429(1)
17.8 Path-Specific Effects for Multiple Mediators
430(2)
17.9 Discussion and Further Challenges
432(7)
18 Search for Causal Models
439(32)
Peter Spirtes
Kun Zhang
18.1 Introduction
439(1)
18.2 Why Causal Search Is Difficult
440(1)
18.3 Assumptions and Terminology
441(3)
18.4 Types of Search
444(1)
18.5 Constraint-Based Search and Hybrid Search
445(11)
18.5.1 Acyclicity, no latent confounders, no selection bias
445(7)
18.5.2 Acyclicity, latent confounders, selection bias
452(2)
18.5.3 Cycles, no latent confounders, no selection bias
454(1)
18.5.4 Cycles, latent confounders, overlapping data sets, experimental and observational data
455(1)
18.6 Score-Based Search
456(6)
18.6.1 Score-based DAG search
456(1)
18.6.2 Score-based equivalence class search
457(1)
18.6.3 Functional causal discovery
458(4)
18.7 Conclusion and Discussions
462(9)
V Applications 471(58)
19 Graphical Models for Forensic Analysis
473(24)
A. Philip Dawid
Julia Mortera
19.1 Introduction
473(1)
19.2 Bayesian Networks for the Analysis of Evidence
474(4)
19.3 Object-Oriented Networks
478(2)
19.3.1 Generic modules
478(2)
19.4 Quantitative Analysis
480(1)
19.5 Bayesian Networks for Forensic Genetics
481(3)
19.5.1 Bayesian networks for simple criminal cases
482(1)
19.5.2 Bayesian network for simple paternity cases
482(2)
19.5.3 Bayesian networks for complex cases
484(1)
19.6 Bayesian Networks for DNA Mixtures
484(5)
19.6.1 Qualitative data
485(1)
19.6.2 Quantitative data
486(2)
19.6.3 Further developments on DNA mixtures
488(1)
19.7 Analysis of Sensitivity to Assumptions on Founder Genes
489(3)
19.7.1 Uncertainty in allele frequencies
489(1)
19.7.2 Heterogeneous reference population
490(2)
19.8 Conclusions
492(1)
Appendix: Genetic Background
492(5)
20 Graphical Models in Molecular Systems Biology
497(16)
Sach Mukherjee
Chris Oates
20.1 Background
498(2)
20.1.1 DNA, RNA and proteins
498(1)
20.1.2 Biological networks
498(1)
20.1.3 A motivating problem
499(1)
20.1.4 Notation
499(1)
20.2 Methods for Snapshot or Static Data
500(4)
20.2.1 Gaussian graphical models
500(2)
20.2.2 Directed acyclic graphs
502(1)
20.2.3 Heterogeneous data and biological context
503(1)
20.3 Biological Dynamics and Models for Time-Varying Data
504(3)
20.3.1 Cellular dynamics
504(1)
20.3.2 Towards linear models
504(1)
20.3.3 Dynamic Bayesian networks
505(1)
20.3.4 Nonlinear models
505(2)
20.4 Causality
507(1)
20.4.1 Causal discovery for biological data
507(1)
20.4.2 Empirical assessment of causal discovery
507(1)
20.5 Perspective and Outlook
508(5)
21 Graphical Models in Genetics, Genomics, and Metagenomics
513(16)
Hongzhe Li
Jing Ma
21.1 Introduction
513(2)
21.1.1 The human interactome
514(1)
21.1.2 Publicly available databases
514(1)
21.1.3 Genetic terminologies
515(1)
21.2 Network-Based Analysis in Genetics
515(2)
21.2.1 Network-assisted analysis in genome-wide association studies
515(1)
21.2.2 Co-expression network-based association analysis of rare variants
516(1)
21.3 Network-Based eQTL and Integrative Genomic Analysis
517(5)
21.3.1 Detection of trans acting genetic effects
518(1)
21.3.2 A causal mediation framework for integration of GWAS and eQTL studies
519(3)
21.4 Network Models in Metagenomics
522(2)
21.4.1 Covariance based on compositional data
522(1)
21.4.2 Microbial community dynamics
523(1)
21.5 Future Directions and Topics
524(5)
Index 529
Marloes Maathuis is Professor of Statistics at ETH Zurich.

Mathias Drton is Professor of Statistics at the University of Copenhagen and the University of Washington.

Steffen Lauritzen is Professor of Statistics at the University of Copenhagen.

Martin Wainwright is Chancellor's Professor at the University of Berkeley.
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