## Bayesian Analysis

### An enriched conjugate prior for Bayesian nonparametric inference

#### Abstract

The precision parameter $\alpha$ plays an important role in the Dirichlet Pro- cess. When assigning a Dirichlet Process prior to the set of probability measures on $\mathbb{R}^k, k \gt 1$, this can be restrictive in the sense that the variability is determined by a single parameter. The aim of this paper is to construct an enrichment foof the Dirichlet Process that is more flexible with respect to the precision parameter yet still conjugate, starting from the notion of enriched conjugate priors, which have been proposed to address an analogous lack of flexibility of standard conjugate priors in a parametric setting. The resulting enriched conjugate prior allows more flexibility in modelling uncertainty on the marginal and conditionals. We describe an enriched urn scheme which characterizes this process and show that it can also be obtained from the stick-breaking representation of the marginal and conditionals. For non atomic base measures, this allows global clustering of the marginal variables and local clustering of the conditional variables. Finally, we consider an application to mixture models that allows for uncertainty between homoskedasticity and heteroskedasticity.

#### Article information

Source
Bayesian Anal. Volume 6, Number 3 (2011), 359-385.

Dates
First available in Project Euclid: 13 June 2012

https://projecteuclid.org/euclid.ba/1339616468

Digital Object Identifier
doi:10.1214/11-BA614

Mathematical Reviews number (MathSciNet)
MR2843536

Zentralblatt MATH identifier
1330.62219

Subjects
Primary: 62F15: Bayesian inference
Secondary: 60G57: Random measures 62G07: Density estimation

#### Citation

Wade, Sara; Mongelluzzo, Silvia; Petrone, Sonia. An enriched conjugate prior for Bayesian nonparametric inference. Bayesian Anal. 6 (2011), no. 3, 359--385. doi:10.1214/11-BA614. https://projecteuclid.org/euclid.ba/1339616468

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