The Annals of Statistics

Nonparametric Binary Regression: A Bayesian Approach

Abstract

The performance of Bayes estimates are studied, under an assumption of conditional exchangeability. More exactly, for each subject in a data set, let $\xi$ be a vector of binary covariates and let $\eta$ be a binary response variable, with $P\{\eta = 1\mid \xi\} = f(\xi)$. Here, $f$ is an unknown function to be estimated from the data; the subjects are independent, and satisfy a natural "balance" condition. Define a prior distribution on $f$ as $\sum_kw_k\pi_k/\sum_kw_k$, where $\pi_k$ is uniform on the set of $f$ which only depend on the first $k$ covariates and $w_k > 0$ for infinitely many $k$. Bayes estimates are consistent at all $f$ if $w_k$ decreases rapidly as $k$ increase. Otherwise, the estimates are inconsistent at $f \equiv 1/2$.

Article information

Source
Ann. Statist. Volume 21, Number 4 (1993), 2108-2137.

Dates
First available: 12 April 2007

http://projecteuclid.org/euclid.aos/1176349413

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176349413

Mathematical Reviews number (MathSciNet)
MR1245784

Zentralblatt MATH identifier
0797.62031

Subjects
Primary: 62A15
Secondary: 62E20: Asymptotic distribution theory

Citation

Diaconis, P.; Freedman, D. A. Nonparametric Binary Regression: A Bayesian Approach. The Annals of Statistics 21 (1993), no. 4, 2108--2137. doi:10.1214/aos/1176349413. http://projecteuclid.org/euclid.aos/1176349413.