Nonparametric Bayesian Discrete Latent Variable Models for ...

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B Mathematical Appendix Multinomial distribution is the distribution of a random variable that can take a countable number of different values. The conjugate prior for the multinomial distribution is the Dirichlet distribution, which is the multivariate generalization of the Beta distribution. The marginal distribution of each πi is beta It follows from the definition of the Dirichlet distribution and the additive property of the gamma distribution that if (π1, . . . , πk) ∼ D(α1, . . . , αk) and r1, . . . , rl are integers such that 0 < r1, . . . , rl = k, then r1 D( π1, 1 r2 πi . . . , r1+1 rl rl−1+1 r1 πi | α1, . . . , αk) ∼ D( α1, 1 r2 αi . . . , r1+1 rl rl−1+1 In particular, the marginal distribution of each πi is Beta αi, ( k 1 αj) − αi . Scale of the Dirichlet distribution The first and second moments of πj are given by; E{πj} = αj α , Var(πi, πj) = αj(α − αj) α 2 (α + 1) , αi), (B.6) where α = K j=1 αj is referred to as the scale of the distribution. The mean of the distribution does not depend on the scale of the parameters, however scale determines the spread. Posterior distribution The posterior distribution of the πj given multinomially distributed data is: p(π1, . . . , πk | x1, . . . , xn) = D(α1 + n1, . . . , αk + nk), where nj is the number of occurrence of the j th event. The parameters of the Dirichlet distribution can be interpreted as the pseudo observations; for larger scale, the distribution will be more concentrated around the mean, thus the prior will have more effect on the posterior. We can marginalize out the multinomial parameters πj using the Dirichlet integral (B.11) and express the probability of the observation directly in terms of the Dirichlet parameters: p(x1, . . . , xn | α1, . . . , αk) = Γ(α) Γ( k l=1 αl + n) k j=1 Γ(αj + nj) Γ(αj) Thus, the conditional distribution for a new observation xn+1 given the previous observations is: p(xn+1 = j | x1, . . . , xn, α1, . . . , αk) = αj + nj k i=1 αi + n . 120
p3 α 1 0.5 0 0 0.5 0.4 0.3 0.2 0.1 0 0.5 p2 1 1 0.5 p1 1 2 3 p3 0 α 1 0.5 0 0 5 4 3 2 1 0 0.5 p2 1 1 0.5 p1 1 2 3 p3 0 1 0.5 0 0 0.5 B.1 Dirichlet Distribution p2 1 1 0.5 p1 1 2 3 Figure B.1: Effect of the scale of the parameters for the Dirichlet distribution. Top row: Samples from Dirichlet distribution with three different parameter settings. Bottom row: Parameter values. The samples lie on the 2-D simplex denoted by the triangle. Note that the relative magnitudes for the distribution parameters are the same, however the scale changes, which effects the spread of the samples. The samples get more concentrated around the mean the higher the scale gets. α 50 40 30 20 10 0 0 121
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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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ch 2 ch 3 ch 1 ch 2 ch 3 ch 4 3.4 D
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3.5 Discussion 3.5 Discussion In th
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4 Indian Buffet Process Models matr
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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: B Mathematical Appendix B.1 Dirichl
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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