Electronic Journal of Statistics

Posterior asymptotics in the supremum $L_{1}$ norm for conditional density estimation

Pierpaolo De Blasi and Stephen G. Walker

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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.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 3219-3246.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62G08: Nonparametric regression

Nonparametric Bayesian inference posterior asymptotics conditional density estimation regression tree model


De Blasi, Pierpaolo; Walker, Stephen G. Posterior asymptotics in the supremum $L_{1}$ norm for conditional density estimation. Electron. J. Statist. 10 (2016), no. 2, 3219--3246. doi:10.1214/16-EJS1191. https://projecteuclid.org/euclid.ejs/1479287219

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