Bernoulli

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

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

Alessandra Guglielmi and Richard L. Tweedie

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1079559463

Mathematical Reviews number (MathSciNet)
MR2002j:62107

Zentralblatt MATH identifier
1005.62073

Keywords
Dirichlet process Markov chain Monte Carlo Markov chains with general state space mean functional rate of convergence

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


Export citation

References

  • [1] Cifarelli, D.M. and Regazzini, E. (1990) Distribution functions of means of a Dirichlet process. Ann. Statist., 18, 429-442. Abstract can also be found in the ISI/STMA publication
  • [2] Corcoran, J. and Tweedie, R.L. (2001) Perfect sampling of Harris recurrent Markov chains. Ann. Appl. Probab. To appear.
  • [3] Feigin, P.D. and Tweedie, R.L. (1989) Linear functionals and Markov chains associated with Dirichlet processes. Math. Proc. Cambridge Philos. Soc., 105, 579-585.
  • [4] Ferguson, T.S. (1973) A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209-230.
  • [5] Florens, J.-P. and Rolin, J.M. (1994) Bayes, bootstrap, moments. Discussion Paper 94.13, Institut de Statistique, Université Catholique de Louvain.
  • [6] Gelfand, A.E. and Kottas, A. (1999) Full nonparametric Bayesian inference for single and multiple sample problems. Technical report, University of Connecticut.
  • [7] Gugliemi, A. (1998) Numerical analysis for the distribution function of the mean of a Dirichlet process. Quaderno IAMI 98.01, CNR-IAMI.
  • [8] Guglielmi, A., Holmes, C.C. and Walker, S.G. (2000) Perfect simulation involving a continuous and unbounded space. Technical Report TR-00-08, Department of Mathematics, Imperial College, London University.
  • [9] Meyn, S.P. and Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Berlin: Springer-Verlag.
  • [10] Muliere, P. and Secchi, P. (1996) Bayesian nonparametric predictive inference and bootstrap techniques. Ann. Inst. Statist. Math., 48, 663-673. Abstract can also be found in the ISI/STMA publication
  • [11] Muliere, P. and Tardella, L. (1998) Approximating distributions of random functionals of Ferguson- Dirichlet priors. Canad. J. Statist., 26, 283-297. Abstract can also be found in the ISI/STMA publication
  • [12] Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge: Cambridge University Press.
  • [13] Regazzini, E., Guglielmi, A. and Di Nunno, G. (2000) Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Quaderno IAMI 00.12, CNR-IAMI. Abstract can also be found in the ISI/STMA publication
  • [14] Roberts, G.O. and Tweedie, R.L. (1999) Bounds on regeneration times and convergence rates for
  • [15] Markov chains. Stochastic Process. Appl., 80, 211-229. Correction (2001): Stochastic Process. Appl., 91, 337-338.
  • [16] Roberts, G.O. and Tweedie, R.L. (2000) Rates of convergence of stochastically monotone stochastic processes. J. Appl. Probab., 37, 359-373. Abstract can also be found in the ISI/STMA publication
  • [17] Rosenthal, J.S. (1995) Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc., 90, 558-566. Abstract can also be found in the ISI/STMA publication
  • [18] SerØing, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • [19] Sethuraman, J. (1994) A constructive definition of Dirichlet priors. Statist. Sinica, 4, 639-650. Abstract can also be found in the ISI/STMA publication
  • [20] Tweedie, R.L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stochastic Process. Appl., 3, 385-403.