Nonparametric Bayesian Discrete Latent Variable Models for ...

More documents

Recommendations

Info

3 Dirichlet Process Mixture Models from the prior G0, resulting in the acceptance ratio: min 1, N − 1 α F (xi | φc ∗ i ) F (xi | φci ) . (3.36) Algorithm 6 Metropolis-Hastings sampling with restricted Gibbs updates for nonconjugate DPM models The state of the Markov chain consists of the indicator variables c = {c1, . . . , cN} and the parameters of the active components Φ = {φ1, . . . , φ K ‡} Repeatedly sample: for all i = 1, . . . , N do if ci is a singleton then Assign ci to one of the active components with probability n−i,k N−1 Evaluate the assignment with acceptance probability given in eq. (3.35) end if if ci is not a singleton then Assign ci to a new component drawing a parameter φci from G0 Evaluate the assignment with acceptance probability given in eq. (3.36) end if end for for all ci that are not singletons do Assigning ci to one of the active components with probability ∝ n−i,k N−1 F (φk) end for for all k = 1, . . . , K ‡ do Update φk by sampling from its conditional posterior, eq. (3.31) end for Gibbs Sampling using Auxiliary Variables (Walker and Damien, 1998) In this section, we summarize another Gibbs sampling algorithm using auxiliary variables, which is rather different from the methods discussed so far. Repeating from Section 3.2.1, we want to sample from the conditional posterior of the parameter given by eq. (3.27), (θi|xi, θ−i) ∝F (xi | θi) α α + n − 1 G0(·) + =qi0G i 0(·) + qijδθj (·), j=i 1 α + n − 1 j=i δθj (·) (3.37) which requires computing the integral qi0 = α F (x|θ)dG0(θ). To avoid this integral which is intractable for non-conjugate models, Walker and Damien (1998) suggest in- 32
3.2 MCMC Inference in Dirichlet Process Mixture Models troducing a latent variable ui and construct the joint distribution of (θi, ui) given by (θi, ui | xi) ∝ I{u < F (xi | θi)} αG0(θi) + j=i δθj (θj) , (3.38) where I{·} is the indicator function. The full conditionals for the Gibbs sampling are given as (ui|θi, xi) ∼ Uniform 0, F (xi | θi) (3.39) and (θi|ui, xi) ∝ αG0(θi)I{u < F (xi | θi)} + j:u
Page 1: Nonparametric Bayesian Discrete Lat
Page 4 and 5: Matrizen mit unendlich vielen Spalt
Page 7 and 8: Contents Zusammenfassung iii Abstra
Page 9: List of Algorithms 1 Gibbs sampling
Page 13 and 14: Notation Matrices are capitalized a
Page 15: Symbol Meaning IBP Z binary latent
Page 18 and 19: 1 Introduction belief in the prior.
Page 20 and 21: 2 Nonparametric Bayesian Analysis b
Page 22 and 23: 2 Nonparametric Bayesian Analysis s
Page 25 and 26: 3 Dirichlet Process Mixture Models
Page 27 and 28: 3.1 The Dirichlet Process the perfo
Page 29 and 30: α G o G θi x i N 3.1 The Dirichle
Page 31 and 32: 15 10 5 −0.5 0 0.5 2 1 G 0 0 −0
Page 33 and 34: increment process with the correspo
Page 35 and 36: 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 47: 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
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
Page 84 and 85: 4 Indian Buffet Process Models matr
Page 86 and 87: 4 Indian Buffet Process Models In t
Page 88 and 89: 4 Indian Buffet Process Models α z
Page 90 and 91: 4 Indian Buffet Process Models α
Page 92 and 93: 4 Indian Buffet Process Models The
Page 94 and 95: 4 Indian Buffet Process Models colu
Page 96 and 97: 4 Indian Buffet Process Models Pois
Page 98 and 99:
4 Indian Buffet Process Models z α
Page 100 and 101:
4 Indian Buffet Process Models For
Page 102 and 103:
4 Indian Buffet Process Models ciat
Page 104 and 105:
4 Indian Buffet Process Models rati
Page 106 and 107:
4 Indian Buffet Process Models repr
Page 108 and 109:
4 Indian Buffet Process Models samp
Page 110 and 111:
4 Indian Buffet Process Models feat
Page 112 and 113:
4 Indian Buffet Process Models Algo
Page 114 and 115:
4 Indian Buffet Process Models mixi
Page 116 and 117:
4 Indian Buffet Process Models Figu
Page 118 and 119:
4 Indian Buffet Process Models pres
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
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
show all

Nonparametric Bayesian Discrete Latent Variable Models for ...

Create successful ePaper yourself

Delete template?

Save as template?