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, 95 (2), 307-323

Kernel Stick-Breaking Processes

Affiliations

Kernel Stick-Breaking Processes

David B Dunson et al. Biometrika.

Abstract

We propose a class of kernel stick-breaking processes for uncountable collections of dependent random probability measures. The process is constructed by first introducing an infinite sequence of random locations. Independent random probability measures and beta-distributed random weights are assigned to each location. Predictor-dependent random probability measures are then constructed by mixing over the locations, with stick-breaking probabilities expressed as a kernel multiplied by the beta weights. Some theoretical properties of the process are described, including a covariate-dependent prediction rule. A retrospective Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using a simulated example and an epidemiological application.

Figures

Fig. 1
Fig. 1
Results for the kernel stick-breaking reference analysis in the simulation example. Estimated conditional response densities are shown for different percentiles of the predictor, including (a) 10th, (b) 25th, (c) 50th, (d) 75th, (e) 90th. The raw data and mean regression estimator are shown in (f). The solid lines are the posterior means, the dashed lines are pointwise 99% credible intervals, and the dotted lines are the true values.
Fig. 2
Fig. 2
dde vs gestational age at delivery in days for 2313 women in the Longnecker et al. (2001) study. The solid line is the conditional predictive mean, while the dotted lines are 99% pointwise credible intervals. Vertical dashed lines are dde quintiles.
Fig. 3
Fig. 3
Estimated densities of gestational age at delivery (in days) conditionally on dde, f(y|x), for the kernel stick-breaking reference analysis. Estimates correspond to different percentiles of the predictor distribution, including (a) 10th, (b) 60th, (c) 90th and (d) 99th. Solid lines represent posterior means, and dashed lines represent 99% credible intervals.
Fig. 4
Fig. 4
Estimated probability gestational age at delivery is less than T weeks versus dde dose, for (a) T = 33, (b) T = 35, (c) T = 37, (d) T = 40. Solid lines are posterior means and dashed lines are pointwise 99% credible intervals.

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