Nonparametric Bayesian Discrete Latent Variable Models for ...

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5 Conclusions has been defined as an extension of DP. An IBP can be used as a nonparametric prior over latent binary features in a hierarchical model. We have described several different approaches for defining and generalizing the distribution given by the IBP. We have summarized the MCMC techniques developed for inference on the IBP models. The modeling performance of an IBP model is demonstrated by successfully learning the features of handwritten digits. We have formulated a nonparametric version of a choice model, elimination by aspects (EBA) and applied the slice sampling algorithm to infer latent features of alternatives in a choice experiment. The development and assessment of sophisticated models using DP and extensions as well as inference techniques for these models is an ongoing area of research. The flexibility and statistical strength of the nonparametric Bayesian models motivates application of these models to challenging real-world problems. 114
A Details of Derivations for the Stick-Breaking Representation of IBP The stick-breaking construction of the IBP has been summarized in Section 4.1.2. Here, we give details of derivation for obtaining this representation. A.1 Densities of ordered stick lengths The density for the kth largest feature presence probability is obtained by considering the decreasing order statistics of K i.i.d. random variables in the limit as K → ∞. We define µl1 ≥ µl2 ≥ · · · ≥ µlK to be the decreasing ordering of µ1, . . . , µK, and Lk to be the set of indices other than the k largest µ’s, Lk = {1, . . . , K}\{l1, . . . , lk}. Thus, the indices for the k largest feature presence probabilities are {l1, . . . , lk}. Denoting µ (k) = µlk , the k largest values among µ1:K is denoted as µ (1:k) = {µ (1), . . . , µ (k)} and we have µl ≤ µ (k) for all l ∈ Lk. (A.1) Thus, the range of the subsequent µl’s given µ (k) is restricted to [0, µ (k)], resulting in the following pdf: p(µl | µ (1:k)) = α K µ(k) 0 µ α K −1 l α α t K K −1 dt = α K µ− α K (k) Integrating, the cdf for µl conditioned on µ (1:k) is found to be F (µl | µ (1:k)) = = µ µl 0 α − K (k) α K α K µ− (k) t α K −1 dt µ α K l I(0 ≤ µl ≤ µ (k)) + I(µ (k) < µl). µ α K −1 l . (A.2) (A.3) 115
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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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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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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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