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4 Indian Buffet Process Models α z µ k Figure 4.2: Graphical representation for the binary feature model using finite binary matrix. The parameters µk determine the presence or absence of the features that define the inputs. To see the connection between the distribution over the binary matrix and the sequential Indian buffet process, we can think of starting with a matrix of zeros and sequentially setting each entry of the matrix. The incremental conditional probabilities are given as: P (zik | zjk, 1 ≤ j < i) = P (zik | µk)P (µk | zjk, 1 ≤ j < i)dµk Γ(α/K + i) = Γ(m.
4.1 The Indian Buffet Process and Ghahramani (2005) show that the distribution over the equivalence classes of the matrices generated by this model and the IBP are the same, which we briefly present below. See the referred paper for details of the derivation. Conditioned on µk, the rows of the matrix are assumed to be independent, therefore the joint distribution over a column is P (zk | µk) = (µk) m.,k (1 − µk) N−m.,k , (4.10) where m.,k = N i=1 zik, is the number of 1s in column k, i.e. the number of objects that share the kth feature. We can integrate over µk using the Dirichlet integral (B.11), and express the joint probability of the column directly in terms of the hyperparameter α: P (zk) = P (zk | µk)P (µk) dµk Γ(α/K + 1) = = α K k=1 Γ(α/K) µα/K−1 k µ m.,k k (1 − µk) N−m.,k dµk Γ(m.,k + α/K)Γ(N − m.,k + 1) . Γ(N + α/K + 1) k=1 (4.11) Assuming the columns to be independent, the distribution over the whole matrix becomes K K α Γ(m.,k + α/K)Γ(N − m.,k + 1) P (Z) = P (zk) = . (4.12) K Γ(N + α/K + 1) Considering the lof -equivalence classes of the matrices, the number of matrices Z defined by the above generative model that map to the equivalent matrix [Z] is K! 2N , −1 h=1 Kh! which leads to the following distribution over [Z], P ([Z]) = K! 2 N −1 h=1 Kh! K k=1 α K Γ(m.,k + α/K)Γ(N − m.,k + 1) . (4.13) Γ(N + α/K + 1) This is the distribution of the equivalence classes for the binary matrix with K < ∞ columns. The distribution over the matrix with infinitely many columns can be obtained by separating the terms for non-zero columns (i.e., columns with m.,k > 0) and the zero columns and simply taking the limit as the number of columns K → ∞. The limiting distribution is found to be equal to eq. (4.2). We repeat the argument of the previous section: even though Z has infinitely many columns, its distribution favors sparsity. We can compute the expected number of nonzero entries in Z for the finite case, and take the limit to obtain the expression for the 73
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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
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
Page 59 and 60: −1 µ y Σy Σy D ξ normal R 3.3
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 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
Page 137 and 138: 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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