## The Annals of Statistics

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

### Nonparametric Binary Regression: A Bayesian Approach

P. Diaconis and D. A. Freedman

#### 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 in Project Euclid: 12 April 2007

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aos/1176349413

**Mathematical Reviews number (MathSciNet)**

MR1245784

**Zentralblatt MATH identifier**

0797.62031

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62A15

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Consistency Bayes estimates model selection binary regression

#### Citation

Diaconis, P.; Freedman, D. A. Nonparametric Binary Regression: A Bayesian Approach. Ann. Statist. 21 (1993), no. 4, 2108--2137. doi:10.1214/aos/1176349413. https://projecteuclid.org/euclid.aos/1176349413