Nonparametric Bayesian Discrete Latent Variable Models for ...

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4 Indian Buffet Process Models mixing time 10 2 10 0 mixing times for α Stick−BreakingSemi−Ordered Gibbs Sampling mixing time 10 2 10 0 mixing times of active features Stick−BreakingSemi−Ordered Gibbs Sampling Figure 4.7: Comparison of the mixing times for α (left) and the number of active features K ‡ (right) for the slice sampler on stick-breaking representation with decreasing stick lengths, the slice sampling for the semi-ordered stick-breaking representation, and for the conjugate Gibbs sampler. As expected, mixing is slowest when the stick lengths are constrained to have the strictly decreasing ordering. The slice sampler on the semi-ordered stick lengths not exploiting conjugacy is as fast as the Gibbs sampler using conjugacy. five auxiliary features for unique feature updates. Since the approximate scheme is more sensitive to hyperparameter values, we fixed the values of α and σA for the initial 25 iterations for both samplers. The mixing time of the approximate scheme is slower as expected, since only a finite number of new features with specific parameters are considered instead of integrating over the infinitely many features. Nevertheless, the modeling performance of the approximate scheme is found to be comparable to the exact scheme. The latent features making up the images and the generating noise scale can be accurately recovered. The trace plots of both samplers are depicted in Figure 4.8 and Figure 4.9. 4.4 A Flexible Infinite Latent Feature Model The binary feature model used in the previous section models the observations as a combination of the latent features. Each latent feature is either present with weight 1 or does not contribute at all to the observation. We can obtain a more powerful model by allowing the latent features to take on real values while preserving sparsity. We use a sparse real-valued latent feature matrix F composed of two components: a binary matrix Z that encodes the presence or absence of the features, and a real-valued matrix Y expressing the weight of each feature for each object, as suggested in Griffiths and Ghahramani (2005). In particular, we define Y to be a matrix of the same size as Z, with i.i.d. zero mean unit variance Gaussian entries. We model each xi as (xi|zi, yi, A, σ 2 x) ∼ N ((zi,: ⊗ yi,:)A, σ 2 xI), (4.62) where ⊗ denotes elementwise multiplication. Specification for the rest of the parameters A, σx, σA and α is as in the previous section. Since the data distribution eq. (4.62) does not depend on the zero columns of Z, the 98
logP(F|X,σ x ) K α σ x σ a −4.3 −4.5 15 5 4 2 1 0.5 0 1.5 1 0.5 10 3 4.4 A Flexible Infinite Latent Feature Model conjugate sampling 250 500 750 1000 250 500 750 1000 250 500 750 1000 250 500 750 1000 250 500 iterations 750 1000 Figure 4.8: Trace plots for the Gibbs sampling using conjugacy. The sampler converges in a few iterations to the high probability regions, employing five to seven latent features. The generating value of σx is recovered. logP(F|X,σ x ) K α σ x σ a −4.3 10 3 non−conjugate sampling −4.5 0 100 200 300 400 500 600 700 800 900 1000 15 5 4 2 1 0.5 0 1.5 1 0.5 250 500 750 1000 250 500 750 1000 250 500 750 1000 250 500 iterations 750 1000 Figure 4.9: Trace plots for the approximate Gibbs sampler not exploiting conjugacy with five auxiliary features for unique feature updates. Compared to the conjugate Gibbs sampler, the chain moves slower, especially for K. However the samples for all parameters have similar values to the samples from the conjugate Gibbs sampler. 99
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Nonparametric Bayesian Discrete Lat
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Matrizen mit unendlich vielen Spalt
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Contents Zusammenfassung iii Abstra
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List of Algorithms 1 Gibbs sampling
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Notation Matrices are capitalized a
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Symbol Meaning IBP Z binary latent
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1 Introduction belief in the prior.
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2 Nonparametric Bayesian Analysis b
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2 Nonparametric Bayesian Analysis s
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3 Dirichlet Process Mixture Models
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3.1 The Dirichlet Process the perfo
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α G o G θi x i N 3.1 The Dirichle
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15 10 5 −0.5 0 0.5 2 1 G 0 0 −0
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increment process with the correspo
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α G o π k c i θk x i 8 N 3.1 The
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3.1 The Dirichlet Process Eq. (3.21
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number of components, K number of c
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3.2 MCMC Inference in Dirichlet Pro
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and Bush and MacEachern (1996). 3.2
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3.2.2 Algorithms for non-Conjugate
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∗ π ∗ π s 3.2 MCMC Inference
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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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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
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Page 79 and 80: ch 2 ch 3 ch 1 ch 2 ch 3 ch 4 3.4 D
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Page 88 and 89: 4 Indian Buffet Process Models α z
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Page 124 and 125: 4 Indian Buffet Process Models P+ P
Page 126 and 127: 4 Indian Buffet Process Models tEBA
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Page 130 and 131: 5 Conclusions has been defined as a
Page 132 and 133: A Details of Derivations for the St
Page 135 and 136: B Mathematical Appendix B.1 Dirichl
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Page 139 and 140: Construction of A Process B.4 Equal
Page 141 and 142: Bibliography D. Aldous. Exchangeabi
Page 143 and 144: Bibliography T. S. Ferguson. Prior
Page 145 and 146: Bibliography L. F. James and J. W.
Page 147 and 148: Bibliography R. M. Neal. Probabilis
Page 149 and 150: Bibliography Y. W. Teh, M. I. Jorda
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