Open Access
2016 Posterior asymptotics in the supremum $L_{1}$ norm for conditional density estimation
Pierpaolo De Blasi, Stephen G. Walker
Electron. J. Statist. 10(2): 3219-3246 (2016). DOI: 10.1214/16-EJS1191


In this paper we study posterior asymptotics for conditional density estimation in the supremum $L_{1}$ norm. Compared to the expected $L_{1}$ norm, the supremum $L_{1}$ norm allows accurate prediction at any designated conditional density. We model the conditional density as a regression tree by defining a data dependent sequence of increasingly finer partitions of the predictor space and by specifying the conditional density to be the same across all predictor values in a partition set. Each conditional density is modeled independently so that the prior specifies a type of dependence between conditional densities which disappears after a certain number of observations have been observed. The rate at which the number of partition sets increases with the sample size determines when the dependence between pairs of conditional densities is set to zero and, ultimately, drives posterior convergence at the true data distribution.


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Pierpaolo De Blasi. Stephen G. Walker. "Posterior asymptotics in the supremum $L_{1}$ norm for conditional density estimation." Electron. J. Statist. 10 (2) 3219 - 3246, 2016.


Received: 1 December 2015; Published: 2016
First available in Project Euclid: 16 November 2016

zbMATH: 1357.62199
MathSciNet: MR3572847
Digital Object Identifier: 10.1214/16-EJS1191

Primary: 62G20
Secondary: 62G08

Keywords: Conditional density estimation , nonparametric Bayesian inference , posterior asymptotics , regression tree model

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.10 • No. 2 • 2016
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