Bernoulli

  • Bernoulli
  • Volume 25, Number 2 (2019), 877-901.

Bayesian consistency for a nonparametric stationary Markov model

Minwoo Chae and Stephen G. Walker

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider posterior consistency for a Markov model with a novel class of nonparametric prior. In this model, the transition density is parameterized via a mixing distribution function. Therefore, the Wasserstein distance between mixing measures can be used to construct neighborhoods of a transition density. The Wasserstein distance is sufficiently strong, for example, if the mixing distributions are compactly supported, it dominates the sup-$L_{1}$ metric. We provide sufficient conditions for posterior consistency with respect to the Wasserstein metric provided that the true transition density is also parametrized via a mixing distribution. In general, when it is not be parameterized by a mixing distribution, we show the posterior distribution is consistent with respect to the average $L_{1}$ metric. Also, we provide a prior whose support is sufficiently large to contain most smooth transition densities.

Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 877-901.

Dates
Received: April 2016
Revised: September 2017
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862838

Digital Object Identifier
doi:10.3150/17-BEJ1007

Mathematical Reviews number (MathSciNet)
MR3920360

Zentralblatt MATH identifier
07049394

Keywords
Kullback–Leibler support mixtures nonparametric Markov model posterior consistency Wasserstein metric

Citation

Chae, Minwoo; Walker, Stephen G. Bayesian consistency for a nonparametric stationary Markov model. Bernoulli 25 (2019), no. 2, 877--901. doi:10.3150/17-BEJ1007. https://projecteuclid.org/euclid.bj/1551862838


Export citation

References

  • [1] Antoniano-Villalobos, I. and Walker, S.G. (2015). Bayesian consistency for Markov models. Sankhya A 77 106–125.
  • [2] Antoniano-Villalobos, I. and Walker, S.G. (2016). A nonparametric model for stationary time series. J. Time Series Anal. 37 126–142.
  • [3] Bissiri, P.G. and Ongaro, A. (2014). On the topological support of species sampling priors. Electron. J. Stat. 8 861–882.
  • [4] Bruni, C. and Koch, G. (1985). Identifiability of continuous mixtures of unknown Gaussian distributions. Ann. Probab. 13 1341–1357.
  • [5] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. J. Econometrics 130 307–335.
  • [6] Chen, X., Wu, W.B. and Yi, Y. (2009). Efficient estimation of copula-based semiparametric Markov models. Ann. Statist. 37 4214–4253.
  • [7] Darsow, W.F., Nguyen, B. and Olsen, E.T. (1992). Copulas and Markov processes. Illinois J. Math. 36 600–642.
  • [8] Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • [9] Ghosal, S. and Tang, Y. (2006). Bayesian consistency for Markov processes. Sankhyā 68 227–239.
  • [10] Gibbs, A.L. and Su, F.E. (2002). On choosing and bounding probability metrics. Int. Stat. Rev. 70 419–435.
  • [11] Glynn, P.W. and Ormoneit, D. (2002). Hoeffding’s inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143–146.
  • [12] Hjort, N. L., Holmes, C., Müller, P. and Walker, S. G. (2010). Bayesian Nonparametrics. Cambridge University Press.
  • [13] Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Boca Raton: CRC Press.
  • [14] Kantorovich, L.V. and Rubinstein, G.Š. (1958). On a space of completely additive functions. Vestnik Leningrad Univ. Math. 13 52–59.
  • [15] Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates. Ann. Statist. 12 351–357.
  • [16] Loève, M. (1963). Probability Theory. 3rd ed. D. Van Nostrand: Princeton, N.J.-Toronto, Ont.-London.
  • [17] Majumdar, S. (1992). On topological support of Dirichlet prior. Statist. Probab. Lett. 15 385–388.
  • [18] Mena, R.H. and Walker, S.G. (2005). Stationary autoregressive models via a Bayesian nonparametric approach. J. Time Series Anal. 26 789–805.
  • [19] Merkle, M. (2000). Topics in weak convergence of probability measures. Zb. Rad. (Beogr.) 9 235–274.
  • [20] Meyn, S.P. and Tweedie, R.L. (2012). Markov Chains and Stochastic Stability. New York: Springer Science & Business Media.
  • [21] Nelsen, R.B. (2003). Properties and applications of copulas: A brief survey. In Proceedings of the First Brazilian Conference on Statistical Modeling in Insurance and Finance (J. Dhaene, N. Kolev and P. Morettin, eds.) 10–28. Sao Paulo: Univ. Press USP.
  • [22] Nguyen, X. (2013). Convergence of latent mixing measures in finite and infinite mixture models. Ann. Statist. 41 370–400.
  • [23] Peel, D. and McLachlan, G.J. (2000). Robust mixture modelling using the t distribution. Stat. Comput. 10 339–348.
  • [24] Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8 229–231.
  • [25] Tallis, G. and Chesson, P. (1982). Identifiability of mixtures. J. Aust. Math. Soc. 32 339–348.
  • [26] Tang, Y. and Ghosal, S. (2007). Posterior consistency of Dirichlet mixtures for estimating a transition density. J. Statist. Plann. Inference 137 1711–1726.
  • [27] Teicher, H. (1960). On the mixture of distributions. Ann. Math. Stat. 31 55–73.
  • [28] Teicher, H. (1963). Identifiability of finite mixtures. Ann. Math. Stat. 34 1265–1269.
  • [29] Teicher, H. et al. (1961). Identifiability of mixtures. Ann. Math. Stat. 32 244–248.
  • [30] Walker, S. (2003). On sufficient conditions for Bayesian consistency. Biometrika 90 482–488.
  • [31] Walker, S. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028–2043.
  • [32] Wu, J., Wang, X. and Walker, S.G. (2014). Bayesian nonparametric inference for a multivariate copula function. Methodol. Comput. Appl. Probab. 16 747–763.
  • [33] Wu, J., Wang, X. and Walker, S.G. (2015). Bayesian nonparametric estimation of a copula. J. Stat. Comput. Simul. 85 103–116.
  • [34] Wu, Y. and Ghosal, S. (2010). The $L_{1}$-consistency of Dirichlet mixtures in multivariate Bayesian density estimation. J. Multivariate Anal. 101 2411–2419.