1. Bayesian kernel mixtures for counts
Antonio Canale & David B. Dunson
Presented by Yingjian Wang
Apr. 29, 2011

2. 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;

3. Modeling of counts Mixture of Poissons: a) Not a nonparametric way;
b) Only accounts for cases where the variance is greater than the mean;

4. 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.

5. 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.

6. Univariate rounded kernel

7. Univariate rounded kernel (2)
Existence:
Consistence: (the mapping g(.) maintains KL neighborhoods.)

8. Examples of rounded kernels
Rounded Gaussian kernel:
Other kernels:
log-normal, gamma, Weibull densities.

9. Eliciting the thresholds

10. A Gibbs sampling algorithm

11. Experiment with univariate model
Two scenarios:
Two standards:
Results:

12. Extension to multivariate model

13. Telecommunication data
Data from 2050 SIM cards, with multivariate:
yi=[yi1, yi2, yi3, yi4, yi5],
Compare the RMG with generalized additive model (GAM):