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E-raamat: Variational Bayesian Learning Theory

(Technische Universität Berlin), , (University of Tokyo)
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 11-Jul-2019
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316997215
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Variational Bayesian Learning TheorySuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 11-Jul-2019
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316997215
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Variational Bayesian learning is one of the most popular methods in machine learning. Designed for researchers and graduate students in machine learning, this book summarizes recent developments in the non-asymptotic and asymptotic theory of variational Bayesian learning and suggests how this theory can be applied in practice. The authors begin by developing a basic framework with a focus on conjugacy, which enables the reader to derive tractable algorithms. Next, it summarizes non-asymptotic theory, which, although limited in application to bilinear models, precisely describes the behavior of the variational Bayesian solution and reveals its sparsity inducing mechanism. Finally, the text summarizes asymptotic theory, which reveals phase transition phenomena depending on the prior setting, thus providing suggestions on how to set hyperparameters for particular purposes. Detailed derivations allow readers to follow along without prior knowledge of the mathematical techniques specific to Bayesian learning.

Designed for researchers and graduate students in machine learning, this book introduces the theory of variational Bayesian learning, a popular machine learning method, and suggests how to make use of it in practice. Detailed derivations allow readers to follow along without prior knowledge of the specific mathematical techniques.

Arvustused

'This book presents a very thorough and useful explanation of classical (pre deep learning) mean field variational Bayes. It covers basic algorithms, detailed derivations for various models (eg matrix factorization, GLMs, GMMs, HMMs), and advanced theory, including results on sparsity of the VB estimator, and asymptotic  properties (generalization bounds).' Kevin Murphy, Research scientist, Google Brain 'This book is an excellent and comprehensive reference on the topic of Variational Bayes (VB) inference, which is heavily used in probabilistic machine learning. It covers VB theory and algorithms, and gives a detailed exploration of these methods for matrix factorization and extensions. It will be an essential guide for those using and developing VB methods.' Chris Williams, University of Edinburgh

Muu info

This introduction to the theory of variational Bayesian learning summarizes recent developments and suggests practical applications.
Preface ix
Nomenclature xii
Part I Formulation
1(60)
1 Bayesian Learning
3(36)
1.1 Framework
3(7)
1.2 Computation
10(29)
2 Variational Bayesian Learning
39(22)
2.1 Framework
39(12)
2.2 Other Approximation Methods
51(10)
Part II Algorithm
61(86)
3 VB Algorithm for Multilinear Models
63(40)
3.1 Matrix Factorization
63(11)
3.2 Matrix Factorization with Missing Entries
74(6)
3.3 Tensor Factorization
80(7)
3.4 Low-Rank Subspace Clustering
87(6)
3.5 Sparse Additive Matrix Factorization
93(10)
4 VB Algorithm for Latent Variable Models
103(29)
4.1 Finite Mixture Models
103(12)
4.2 Other Latent Variable Models
115(17)
5 VB Algorithm under No Conjugacy
132(15)
5.1 Logistic Regression
132(3)
5.2 Sparsity-Inducing Prior
135(2)
5.3 Unified Approach by Local VB Bounds
137(10)
Part III Nonasymptotic Theory
147(192)
6 Global VB Solution of Fully Observed Matrix Factorization
149(35)
6.1 Problem Description
150(2)
6.2 Conditions for VB Solutions
152(1)
6.3 Irrelevant Degrees of Freedom
153(4)
6.4 Proof of Theorem 6.4
157(3)
6.5 Problem Decomposition
160(2)
6.6 Analytic Form of Global VB Solution
162(1)
6.7 Proofs of Theorem 6.7 and Corollary 6.8
163(8)
6.8 Analytic Form of Global Empirical VB Solution
171(2)
6.9 Proof of Theorem 6.13
173(7)
6.10 Summary of Intermediate Results
180(4)
7 Model-Induced Regularization and Sparsity Inducing Mechanism
184(21)
7.1 VB Solutions for Special Cases
184(3)
7.2 Posteriors and Estimators in a One-Dimensional Case
187(8)
7.3 Model-Induced Regularization
195(7)
7.4 Phase Transition in VB Learning
202(2)
7.5 Factorization as ARD Model
204(1)
8 Performance Analysis of VB Matrix Factorization
205(31)
8.1 Objective Function for Noise Variance Estimation
205(2)
8.2 Bounds of Noise Variance Estimator
207(2)
8.3 Proofs of Theorem 8.2 and Corollary 8.3
209(5)
8.4 Performance Analysis
214(14)
8.5 Numerical Verification
228(2)
8.6 Comparison with Laplace Approximation
230(2)
8.7 Optimality in Large-Scale Limit
232(4)
9 Global Solver for Matrix Factorization
236(19)
9.1 Global VB Solver for Fully Observed MF
236(2)
9.2 Global EVB Solver for Fully Observed MF
238(4)
9.3 Empirical Comparison with the Standard VB Algorithm
242(5)
9.4 Extension to Nonconjugate MF with Missing Entries
247(8)
10 Global Solver for Low-Rank Subspace Clustering
255(24)
10.1 Problem Description
255(3)
10.2 Conditions for VB Solutions
258(1)
10.3 Irrelevant Degrees of Freedom
259(1)
10.4 Proof of Theorem 10.2
259(5)
10.5 Exact Global VB Solver (EGVBS)
264(3)
10.6 Approximate Global VB Solver (AGVBS)
267(3)
10.7 Proof of Theorem 10.7
270(4)
10.8 Empirical Evaluation
274(5)
11 Efficient Solver for Sparse Additive Matrix Factorization
279(15)
11.1 Problem Description
279(3)
11.2 Efficient Algorithm for SAMF
282(2)
11.3 Experimental Results
284(10)
12 MAP and Partially Bayesian Learning
294(45)
12.1 Theoretical Analysis in Fully Observed MF
295(34)
12.2 More General Cases
329(3)
12.3 Experimental Results
332(7)
Part IV Asymptotic Theory
339(177)
13 Asymptotic Learning Theory
341(44)
13.1 Statistical Learning Machines
341(3)
13.2 Basic Tools for Asymptotic Analysis
344(2)
13.3 Target Quantities
346(5)
13.4 Asymptotic Learning Theory for Regular Models
351(15)
13.5 Asymptotic Learning Theory for Singular Models
366(16)
13.6 Asymptotic Learning Theory for VB Learning
382(3)
14 Asymptotic VB Theory of Reduced Rank Regression
385(44)
14.1 Reduced Rank Regression
385(11)
14.2 Generalization Properties
396(30)
14.3 Insights into VB Learning
426(3)
15 Asymptotic VB Theory of Mixture Models
429(26)
15.1 Basic Lemmas
429(5)
15.2 Mixture of Gaussians
434(9)
15.3 Mixture of Exponential Family Distributions
443(8)
15.4 Mixture of Bernoulli with Deterministic Components
451(4)
16 Asymptotic VB Theory of Other Latent Variable Models
455(45)
16.1 Bayesian Networks
455(6)
16.2 Hidden Markov Models
461(5)
16.3 Probabilistic Context-Free Grammar
466(4)
16.4 Latent Dirichlet Allocation
470(30)
17 Unified Theory for Latent Variable Models
500(16)
17.1 Local Latent Variable Model
500(4)
17.2 Asymptotic Upper-Bound for VB Free Energy
504(3)
17.3 Example: Average VB Free Energy of Gaussian Mixture Model
507(4)
17.4 Free Energy and Generalization Error
511(2)
17.5 Relation to Other Analyses
513(3)
Appendix A James---Stein Estimator 516(4)
Appendix B Metric in Parameter Space 520(5)
Appendix C Detailed Description of Overlap Method 525(2)
Appendix D Optimality of Bayesian Learning 527(2)
Bibliography 529(11)
Subject Index 540
Shinichi Nakajima is a senior researcher at Technische Universität Berlin. His research interests include the theory and applications of machine learning, and he has published papers at numerous conferences and in journals such as the Journal of Machine Learning Research, the Machine Learning Journal, Neural Computation, and IEEE Transactions on Signal Processing. He currently serves as an area chair for NIPS and an action Editor for Digital Signal Processing. Kazuho Watanabe is a lecturer at Toyohashi University of Technology. His research interests include statistical machine learning and information theory, and he has published papers at numerous conferences and in journals such as the Journal of Machine Learning Research, the Machine Learning Journal, IEEE Transactions on Information Theory, and IEEE Transactions on Neural Networks and Learning Systems. Masashi Sugiyama is Director of the RIKEN Center for Advanced Intelligence Project and Professor of Complexity Science and Engineering at the University of Tokyo. His research interests include the theory, algorithms, and applications of machine learning. He has written several books on machine learning, including Density Ratio Estimation in Machine Learning (Cambridge, 2012). He served as program co-chair and general co-chair of the NIPS conference in 2015 and 2016, respectively, and received the Japan Academy Medal in 2017.
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