Nonparametric Bayesian Discrete Latent Variable Models for ...

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3 Dirichlet Process Mixture Models −1 µ y 1 1 Σ D D ξ Σ y normal ρ ρS gamma w µ S y normal 1 β β−D+1 w −1 Wishart inv gamma k k normal Wishart x i N K hyperpriors hyperparameters mixture parameters observations Figure 3.10: Graphical representation of the layered structure of the hierarchical priors in the MoG model with conjugate priors. Note the dependency of the distribution of the component mean on the component precision. Variables are labelled below by the name of their distribution, and the parameters of these distributions are given above. particular value. The graphical representation of the hierarchical model is depicted in Figure 3.10. Note in eq. (3.57) that the precision for the mean is a multiple of the component precision itself. This dependency is probably not generally desirable, but is an unavoidable consequence of requiring conjugacy. Conditionally Conjugate DPMoG If we remove the undesired dependency of the mean on the precision, we no longer have conjugacy. For a more realistic model, we define the prior on µ j to be p(µ j|ξ, R) ∼ N (ξ, R −1 ) (3.60) whose mean vector ξ and precision matrix R are hyperparameters common to all mixture components. Keeping the Wishart prior over the precisions as in eq. (3.58), we obtain the conditionally conjugate model. That is, the prior of the mean is conjugate to the likelihood conditional on S and the prior of the precision is conjugate conditional on µ. See Figure 3.11 for the graphical representation of the hierarchical model. We put hyperpriors on the hyperparameters in both prior specifications to have a robust model. We use the hierarchical model specification of Rasmussen (2000) for the conditionally conjugate model, and a similar specification for the conjugate case. Vague priors are given to the hyperparameters, some of which depend on the observations which technically they ought not to. However, only the empirical mean µ x and the covariance Σx of the data are used in such a way that the full procedure becomes invariant to 42
−1 µ y Σy Σy D ξ normal R 3.3 Empirical Study on the Choice of the Base Distribution Σy w D D −1 β 1 β−D+1 w Wishart −1 Wishart inv gamma µ S k k normal Wishart x i normal N K hyperpriors hyperparameters mixture parameters observations Figure 3.11: Graphical representation of the layered structure of the hierarchical priors in the conditionally conjugate model. The distribution of the component means is independent from the component precision. translations, rotations and rescaling of the data. One could equivalently use unit priors and scale the data before the analysis since finding the overall mean and covariance of the data is not the primary concern of the analysis, rather, we wish to find structure within the data. In detail, for both models the hyperparameters W and β associated with the component precisions Sj are given the following priors, keeping in mind that β should be greater than D−1: W ∼ W(D, 1 D Σx), 1 1 ( β−D+1 ) ∼ G(1, D ). (3.61) For the conjugate model, the priors for the hyperparameters ξ and ρ associated with the mixture means µ j are Gaussian and gamma: ξ ∼ N (µ x, Σx), ρ ∼ G(1/2, 1/2). (3.62) For the non-conjugate case, the component means have a mean vector ξ and a precision matrix R as hyperparameters. We put a Gaussian prior on ξ and a Wishart prior on the precision matrix: ξ ∼ N (µ x, Σx), R ∼ W D, (DΣx) −1 . (3.63) 3.3.2 Inference Using Gibbs Sampling In this section, we describe the MCMC algorithms we utilize for inference on the models. The Markov chain relies on Gibbs updates, where each variable is updated in turn by sampling from its posterior distribution conditional on all other variables. We repeatedly sample the parameters, hyperparameters and the indicator variables from their 43
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Nonparametric Bayesian Discrete Lat
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Matrizen mit unendlich vielen Spalt
Page 7 and 8: Contents Zusammenfassung iii Abstra
Page 9: List of Algorithms 1 Gibbs sampling
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: model can be written in the form of
Page 61 and 62: the log likelihood term is: where a
Page 63 and 64: 3.3 Empirical Study on the Choice o
Page 65 and 66: autocovariance coefficient 1 0.8 0.
Page 67 and 68: # of data points # of data points 5
Page 69 and 70: 3.4 Dirichlet Process Mixtures of F
Page 71 and 72: µ y Σy ξ R 0 ν w normal µ −1
Page 73 and 74: ch1 ch2 ch3 ch4 3.4 Dirichlet Proce
Page 75 and 76: 3.4 Dirichlet Process Mixtures of F
Page 77 and 78: # of components # of components # o
Page 79 and 80: ch 2 ch 3 ch 1 ch 2 ch 3 ch 4 3.4 D
Page 81: 3.5 Discussion 3.5 Discussion In th
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Page 86 and 87: 4 Indian Buffet Process Models In t
Page 88 and 89: 4 Indian Buffet Process Models α z
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Page 92 and 93: 4 Indian Buffet Process Models The
Page 94 and 95: 4 Indian Buffet Process Models colu
Page 96 and 97: 4 Indian Buffet Process Models Pois
Page 98 and 99: 4 Indian Buffet Process Models z α
Page 100 and 101: 4 Indian Buffet Process Models For
Page 102 and 103: 4 Indian Buffet Process Models ciat
Page 104 and 105: 4 Indian Buffet Process Models rati
Page 106 and 107: 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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