nips nips2008 nips2008-236 nips2008-236-reference knowledge-graph by maker-knowledge-mining
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Author: Daniel M. Roy, Yee W. Teh
Abstract: We describe a novel class of distributions, called Mondrian processes, which can be interpreted as probability distributions over kd-tree data structures. Mondrian processes are multidimensional generalizations of Poisson processes and this connection allows us to construct multidimensional generalizations of the stickbreaking process described by Sethuraman (1994), recovering the Dirichlet process in one dimension. After introducing the Aldous-Hoover representation for jointly and separately exchangeable arrays, we show how the process can be used as a nonparametric prior distribution in Bayesian models of relational data. 1
D. J. Aldous. Representations for Partially Exchangeable Arrays of Random Variables. Journal of Multivariate Analysis, 11:581–598, 1981. J. M. Bernardo and A. F. M. Smith. Bayesian theory. John Wiley & Sons, 1994. B. de Finetti. Funzione caratteristica di un fenomeno aleatorio. Atti della R. Academia Nazionale dei Lincei, Serie 6. Memorie, Classe di Scienze Fisiche, Mathematice e Naturale, 4:251299, 1931. P. Diaconis and S. Janson. Graph limits and exchangeable random graphs. arXiv:0712.2749v1, 2007. P. W. Holland, K. B. Laskey, and S. Leinhardt. Stochastic blockmodels: First steps. Social Networks, 5(2):109 – 137, 1983. D. Hoover. Relations on probability spaces and arrays of random variables. Technical report, Preprint, Institute for Advanced Study, Princeton, NJ, 1979. O. Kallenberg. Probabilistic Symmetries and Invariance Principles. Springer, 2005. C. Kemp, J. Tenenbaum, T. Griffiths, T. Yamada, and N. Ueda. Learning systems of concepts with an infinite relational model. In Proceedings of the 21st National Conference on Artificial Intelligence, 2006. J. F. C. Kingman. On the genealogy of large populations. Journal of Applied Probability, 19:27–43, 1982a. J. F. C. Kingman. The coalescent. Stochastic Processes and their Applications, 13:235–248, 1982b. L. Lov´ sz and B. Szegedy. Limits of dense graph sequences. J. Comb. Theory B, 96:933957, 2006. a K. Nowicki and T. A. B. Snijders. Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, 96:1077–1087(11), 2001. D. M. Roy, C. Kemp, V. Mansinghka, and J. B. Tenenbaum. Learning annotated hierarchies from relational data. In Advances in Neural Information Processing Systems 19, 2007. C. Ryll-Nardzewski. On stationary sequences of random variables and the de Finetti’s equivalence. Colloq. Math., 4:149–156, 1957. J. Sethuraman. A Constructive definition of Dirichlet priors. Statistica Sinica, 4:639–650, 1994. Y. W. Teh, D. G¨ r¨ r, and Z. Ghahramani. Stick-breaking construction for the Indian buffet process. In Proou ceedings of the International Conference on Artificial Intelligence and Statistics, volume 11, 2007. S. Wasserman and C. Anderson. Stochastic a posteriori blockmodels: Construction and assessment. Social Networks, 9(1):1 – 36, 1987. S. Wasserman and K. Faust. Social Network Analysis: Methods and Applications, pages 64–65. Cambridge University Press, 1994. Z. Xu, V. Tresp, K. Yu, and H.-P. Kriegel. Infinite Hidden Relational Models. In Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence, 2006. 8