## Bernoulli

• Bernoulli
• Volume 7, Number 4 (2001), 573-592.

### Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process

#### Abstract

The distribution $\mathcal{M}_\alpha$ of the mean $\Gamma_\alpha$ of a Dirichlet process on the real line, with parameter $\alpha$, can be characterized as the invariant distribution of a real Markov chain $\Gamma_n$. In this paper we prove that, if $\alpha$ has finite expectation, the rate of convergence (in total variation) of $\Gamma_n$ to $\Gamma_\alpha$ is geometric. Upper bounds on the rate of convergence are found which seem effective, especially in the case where α has a support which is not doubly infinite. We use this to study an approximation procedure for $\mathcal{M}_\alpha$, and evaluate the approximation error in simulating $\mathcal{M}_\alpha$ using this chain. We include examples for a comparison with some of the existing procedures for approximating $\mathcal{M}_\alpha$, and show that the Markov chain approximation compares well with other methods.

#### Article information

Source
Bernoulli, Volume 7, Number 4 (2001), 573-592.

Dates
First available in Project Euclid: 17 March 2004

https://projecteuclid.org/euclid.bj/1079559463

Mathematical Reviews number (MathSciNet)
MR2002j:62107

Zentralblatt MATH identifier
1005.62073

#### Citation

Guglielmi, Alessandra; Tweedie, Richard L. Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process. Bernoulli 7 (2001), no. 4, 573--592. https://projecteuclid.org/euclid.bj/1079559463

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