Sparse Inverse Covariance Estimation with Graphical LASSO

Sparse Inverse Covariance Estimation with Graphical LASSO

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

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