Nonparametric Bayesian Discrete Latent Variable Models for ...

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Contents 4 Indian Buffet Process Models 67 4.1 The Indian Buffet Process . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.1 A Distribution on Infinite Binary Matrices . . . . . . . . . . . . . 71 4.1.2 Stick-Breaking Construction . . . . . . . . . . . . . . . . . . . . . 74 4.1.3 A Special Case of the Beta Process . . . . . . . . . . . . . . . . 78 4.1.4 Properties of the Distribution on the Infinite Binary Martices . . 79 4.2 MCMC Sampling algorithms for IBLF models . . . . . . . . . . . . . . . 81 4.2.1 Gibbs Sampling for Conjugate IBLF models . . . . . . . . . . . . 83 4.2.2 Approximate Gibbs Sampling for Non-Conjugate Models . . . . 85 4.2.3 Metropolis-Hastings Sampling . . . . . . . . . . . . . . . . . . . . 86 4.2.4 Gibbs Sampling for the Truncated Stick-Breaking Construction . 88 4.2.5 Slice Sampling Using the Stick Breaking Construction . . . . . . 90 4.2.6 Slice Sampling for the Semi-Ordered Stick-Breaking . . . . . . . 93 4.3 Comparing Performances of the Samplers . . . . . . . . . . . . . . . . . 95 4.4 A Flexible Infinite Latent Feature Model . . . . . . . . . . . . . . . . . . 98 4.5 A Choice Model with Infinitely Many Latent Features . . . . . . . . . . 100 4.5.1 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.5.2 Inference using MCMC . . . . . . . . . . . . . . . . . . . . . . . 105 4.5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Conclusions 113 A Details of Derivations for the Stick-Breaking Representation of IBP 115 A.1 Densities of ordered stick lengths . . . . . . . . . . . . . . . . . . . . . . 115 A.2 Probability of a part of Z being inactive . . . . . . . . . . . . . . . . . . 116 B Mathematical Appendix 119 B.1 Dirichlet Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 B.2 Parameterization of the Distributions Used . . . . . . . . . . . . . . . . 122 B.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B.4 Equalities and Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Bibliography 125 viii
List of Algorithms 1 Gibbs sampling for conjugate DPM models with full parameter representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Gibbs sampling for conjugate DPM models using indicator variables and component parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Gibbs sampling for conjugate DPM models using indicator variables without parameter representations. . . . . . . . . . . . . . . . . . . . . . . . 28 4 The No-Gaps algorithm for non-conjugate DPM models. . . . . . . . . . 29 5 Gibbs sampling for non-conjugate DPM models using auxiliary components 31 6 Metropolis-Hastings sampling with restricted Gibbs updates for nonconjugate DPM models . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Auxiliary variable sampling for non-conjugate DPM models. . . . . . . 33 8 Gibbs sampling for truncated DP . . . . . . . . . . . . . . . . . . . . . . 35 9 Retrospective sampling for DPM models using stick-breaking construction 37 10 Slice sampling algorithm for the DPM model . . . . . . . . . . . . . . . 39 11 Gibbs sampling for conjugate IBP . . . . . . . . . . . . . . . . . . . . . 85 12 Approximate Gibbs sampling for non-conjugate IBP . . . . . . . . . . . 87 13 Metropolis-Hastings sampling for IBP . . . . . . . . . . . . . . . . . . . 89 14 Gibbs sampling for truncated IBP . . . . . . . . . . . . . . . . . . . . . 90 15 Slice sampling for stick-breaking IBP . . . . . . . . . . . . . . . . . . . . 93 16 Slice sampling for the semi-ordered IBP . . . . . . . . . . . . . . . . . . 96 ix
Page 1: Nonparametric Bayesian Discrete Lat
Page 4 and 5: Matrizen mit unendlich vielen Spalt
Page 7: Contents Zusammenfassung iii Abstra
Page 13 and 14: Notation Matrices are capitalized a
Page 15: Symbol Meaning IBP Z binary latent
Page 18 and 19: 1 Introduction belief in the prior.
Page 20 and 21: 2 Nonparametric Bayesian Analysis b
Page 22 and 23: 2 Nonparametric Bayesian Analysis s
Page 25 and 26: 3 Dirichlet Process Mixture Models
Page 27 and 28: 3.1 The Dirichlet Process the perfo
Page 29 and 30: α G o G θi x i N 3.1 The Dirichle
Page 31 and 32: 15 10 5 −0.5 0 0.5 2 1 G 0 0 −0
Page 33 and 34: increment process with the correspo
Page 35 and 36: α G o π k c i θk x i 8 N 3.1 The
Page 37 and 38: 3.1 The Dirichlet Process Eq. (3.21
Page 39 and 40: number of components, K number of c
Page 41 and 42: 3.2 MCMC Inference in Dirichlet Pro
Page 43 and 44: and Bush and MacEachern (1996). 3.2
Page 45 and 46: 3.2.2 Algorithms for non-Conjugate
Page 55 and 56: ∗ π ∗ π s 3.2 MCMC Inference
Page 57 and 58: model can be written in the form of
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−1 µ y Σy Σy D ξ normal R 3.3
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the log likelihood term is: where a
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3.3 Empirical Study on the Choice o
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autocovariance coefficient 1 0.8 0.
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# of data points # of data points 5
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3.4 Dirichlet Process Mixtures of F
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µ y Σy ξ R 0 ν w normal µ −1
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ch1 ch2 ch3 ch4 3.4 Dirichlet Proce
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3.4 Dirichlet Process Mixtures of F
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# of components # of components # o
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ch 2 ch 3 ch 1 ch 2 ch 3 ch 4 3.4 D
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3.5 Discussion 3.5 Discussion In th
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4 Indian Buffet Process Models matr
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4 Indian Buffet Process Models In t
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4 Indian Buffet Process Models α z
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4 Indian Buffet Process Models α
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4 Indian Buffet Process Models The
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4 Indian Buffet Process Models colu
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4 Indian Buffet Process Models Pois
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4 Indian Buffet Process Models z α
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4 Indian Buffet Process Models For
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4 Indian Buffet Process Models ciat
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4 Indian Buffet Process Models rati
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4 Indian Buffet Process Models repr
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4 Indian Buffet Process Models samp
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4 Indian Buffet Process Models feat
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4 Indian Buffet Process Models Algo
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4 Indian Buffet Process Models mixi
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4 Indian Buffet Process Models Figu
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4 Indian Buffet Process Models pres
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4 Indian Buffet Process Models ε
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4 Indian Buffet Process Models LL P
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4 Indian Buffet Process Models P+ P
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4 Indian Buffet Process Models tEBA
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4 Indian Buffet Process Models dist
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5 Conclusions has been defined as a
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A Details of Derivations for the St
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B Mathematical Appendix B.1 Dirichl
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p3 α 1 0.5 0 0 0.5 0.4 0.3 0.2 0.1
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Construction of A Process B.4 Equal
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Bibliography D. Aldous. Exchangeabi
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Bibliography T. S. Ferguson. Prior
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Bibliography L. F. James and J. W.
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Bibliography R. M. Neal. Probabilis
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Bibliography Y. W. Teh, M. I. Jorda
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