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4 Indian Buffet Process Models LL P+ P R R+ −300 −350 0 250 500 750 1000 K+ α 8 4 0 0 250 500 750 1000 5 P $ R K+ 3 6 9 0 2 4 6 0 0 250 500 iteration 750 1000 Figure 4.14: Feature matrix representation and simulation results on the toy data: the choice between trips to Paris and Rome. Top left: Features weighted by the associated values are shown. Rows correspond to the alternatives and columns correspond to the features. Darker means smaller in amplitude. The alternatives P and P + share the feature of being the trip to Paris and R and R+ share the feature of being the trip to Rome. P + and R+ share the small bonus denoted by the $ column. 106 α
4.5 A Choice Model with Infinitely Many Latent Features Table 4.1: Choice probabilities for the Paris-Rome example. Columns are chosen over rows. P+ P R R+ P+ 0.50 0 0.48 0.50 P 1 0.50 0.50 0.52 R 0.52 0.50 0.50 1 R+ 0.50 0.48 0 0.50 Consider the choice between two trips that are on offer at a travel agency: one to Paris (P ) and one to Rome (R). Another travel agency offers exactly the same trips to Paris and to Rome except that there is an additional small bonus. We denote these options by P + and R+, respectively. The options in the choice set consist of P, P +, R, R+. We assume that the decision maker assigns the same value to the trip to Paris and to the trip to Rome. Hence, she will be equally likely to prefer the trip to either city. We denote the probability that Paris is chosen over Rome with P (P, R) = 0.5. As it is always better to get a bonus than to not get it we can assume that when given the options P and P + she would choose P + with certainty, i.e. P (P +, P ) = 1. However, since a small bonus will not influence the choice between the two cities, P (P +, R) will be close to 0.5, and likewise for P (R+, P ). We assume that the trip itself to either city has a feature with value w and the bonus is worth 0.01w. Thus, the binary feature matrix weighted with the importance of the features can be represented as shown in the top left of Figure 4.14. The alternatives P and P + share the feature of being the trip to Paris and R and R+ share the feature of being the trip to Rome. P + and R+ share the small bonus denoted by the $ column. Note that there might be some features that trips to either city possess such as taking the plane, going to Europe, etc. Since these features will be common to all options in the choice set they do not affect the choice probabilities. The matrix of the deterministic choice probabilities calculated assuming the EBA model using eq. (4.63) is shown in Table 4.1. We generated choice data of 100 comparisons for each pair using these probabilities and used the EBA model with infinitely many latent features (iEBA) to infer the latent structure of the options. The results of a sample run for the iEBA model are shown in Figure 4.14. The top row shows the feature matrix that is used to generate the data, histogram of the frequencies for the active features and the posterior distribution of the IBP parameter α. The plots below show the change in the log likelihood, the number of active features of the IBP part of the feature matrix and α over the sampling iterations. It can be seen from the trace plots that the burn-in time is very short and the chain seems to mix well. The iEBA model takes only the choice data as input, and it can recover the feature matrices given in Figure 4.14 from the data. Some sample feature matrices from the chain are depicted in Figure 4.15 which shows that the model successfully finds the latent feature representation that was used to generate the data. Note that the mapping from the choice probabilities to features is not unique, that is, several latent feature 107
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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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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 120 and 121: 4 Indian Buffet Process Models ε
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
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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