## Bernoulli

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

### Bayesian consistency for a nonparametric stationary Markov model

#### 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
Revised: September 2017
First available in Project Euclid: 6 March 2019

https://projecteuclid.org/euclid.bj/1551862838

Digital Object Identifier
doi:10.3150/17-BEJ1007

Mathematical Reviews number (MathSciNet)
MR3920360

Zentralblatt MATH identifier
07049394

#### 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

#### 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.