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Bayesian kernel mixtures for counts Antonio Canale & David B. Dunson Presented by Yingjian Wang Apr. 29, 2011 Outline • Existed models for counts and their drawbacks; • Univariate rounded kernel mixture priors; • Simulation of the univariate model; • Multivariate rounded kernel mixture priors; • Experiment with the multivariate model; Modeling of counts • Mixture of Poissons: a) Not a nonparametric way; b) Only accounts for cases where the variance is greater than the mean; Modeling of counts (2) • DP mixture of Poissons/Multinomial kernel: a) It is non-parametric but, still has the problem of not suitable for under-disperse cases; b) If with multinomial kernel, the dimension of the probability vector is equal to the number of support points, causes overfitting. 4 Modeling of counts (3) • DP with Poisson base measure: a) There is no allowance for smooth deviations from the base; • Motivation: The continuous densities can be accurately approximated using Gaussian kernels. • Idea: Use kernels induced through rounding of continuous kernels. 5 Univariate rounded kernel discrete : y ~ p g () h () continuous: y* ~ f 6 Univariate rounded kernel (2) • Existence: • Consistence: (the mapping g(.) maintains KL neighborhoods.) 7 Examples of rounded kernels • Rounded Gaussian kernel: • Other kernels: log-normal, gamma, Weibull densities. 8 Eliciting the thresholds 9 A Gibbs sampling algorithm 10 Experiment with univariate model • Two scenarios: • Two standards: • Results: 11 Extension to multivariate model 12 Telecommunication data • Data from 2050 SIM cards, with multivariate: yi=[yi1, yi2, yi3, yi4, yi5], Compare the RMG with generalized additive model (GAM): 13