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4 Indian Buffet Process Models column k in row i, given µ (k). This does not depend on i due to the exchangeability of the rows. The distribution of the unordered feature presence probabilities µl after the kth largest one is given by eq. (A.2). The entries zil are Bernoulli distributed with probability µl. Marginalizing over µl, we have; p(zil | µ (k)) = = µ(k) 0 µ(k) 0 p(zil | µl)p(µl | µ (k))dµl α = α α + K µ (k). µl K µ−α/K (k) µ α/K−1 l dµl (4.28) Taking the limit as K → ∞ of the Bernoulli trials with the above probability results in a Poisson(αµ (k)) distribution over the number of non-zero entries in the ith row. This result is intuitive, stating that the expected number of non-zero entries depends on the limiting µ (k) value, and the parameter α. If we consider the entries of the whole row, i.e. the case k = 0, we recover the distribution derived in the previous section for the total number of non-zero entries in a row, Poisson(α). We can also calculate the probability of all entries to the right of column k being zero: p(Z (:,.>k) = 0 | µ (k)) = exp − αHN + α N i=1 (1 − µ (k)) i i , (4.29) where HN is the Nth harmonic number. See Appendix A for the details of derivation. 4.1.3 A Special Case of the Beta Process Beta process has been defined for use in survival analysis as a cumulative hazard rate process with nonnegative independent increments 2 by Hjort (1990). Thibaux and Jordan (2007) show that a special case of the beta process is related to the Indian buffet process (IBP) the way the Dirichlet process is related to the Chinese restaurant process. They define the following generative model for the binary feature matrix Z: A ∼ BP(c, αA0), Z ∼ BeP(A), (4.30) where BeP(A) denotes a Bernoulli process with hazard measure A and BP(c, αA0) denotes a beta process with base measure αA0 and concentration function c. A0 is a probability measure, α a positive scalar, and c a deterministic function. The distribution of Z is equivalent to that of the matrix generated by the IBP(α) for the particular choice of c being the constant function c = 1, BP(1, αA0) . Considering the beta process BP(c, αA0) with a constant concentration function in the above generative model results in a two-parameter generalization of the Indian buffet 2 See Appendix B.3 for the definition of the independent increment processes 78
4.1 The Indian Buffet Process process. Customer i samples the previously sampled dishes with Bernoulli probability m·
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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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Page 45 and 46: 3.2.2 Algorithms for non-Conjugate
Page 47 and 48: 3.2 MCMC Inference in Dirichlet Pro
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
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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 120 and 121: 4 Indian Buffet Process Models ε
Page 122 and 123: 4 Indian Buffet Process Models LL P
Page 124 and 125: 4 Indian Buffet Process Models P+ P
Page 126 and 127: 4 Indian Buffet Process Models tEBA
Page 128 and 129: 4 Indian Buffet Process Models dist
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
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
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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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