Bayesian kernel mixtures for counts

Antonio Canale & David B. DunsonPresented by Yingjian Wang

Apr. 29, 2011

• 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;

Outline

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.

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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.

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Univariate rounded kernel

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*

discrete : ~

( ) ( )

continuous: ~

y p

g h

y f

Univariate rounded kernel (2)

• Existence:

• Consistence: (the mapping g(.) maintains KL neighborhoods.)

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Examples of rounded kernels

• Rounded Gaussian kernel:

• Other kernels: log-normal, gamma, Weibull densities.

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Eliciting the thresholds

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A Gibbs sampling algorithm

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Experiment with univariate model• Two scenarios:

• Two standards:• Results:

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Extension to multivariate model

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

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