logo AFST
Computable upper bounds on the distance to stationarity for Jovanovski and Madras’s Gibbs sampler
;
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, pp. 935-947.

An upper bound on the Wasserstein distance to stationarity is developed for a class of Markov chains on . This result, which is a generalization of Diaconis et al.’s (2009) Theorem 2.2, is applied to a Gibbs sampler Markov chain that was introduced and analyzed by Jovanovski and Madras (2014). The resulting Wasserstein bound is converted into a total variation bound (using results from Madras and Sezer (2010)), and the total variation bound is compared to an alternative bound derived by Jovanovski and Madras (2014).

Une borne supérieure est obtenue pour la distance de Wasserstein à la stationnarité pour une classe de chaînes de Markov sur . Ce résultat, qui est une généralisation du théorème 2.2 de Diaconis et al. (2009), est appliqué à l’échantillonneur de Gibbs introduit et analysé par Jovanovski et Madras (2014). La borne de Wasserstein qui en résulte est transformée en une borne en variation totale (en utilisant des résultats de Madras et Sezer (2010)), qui est ensuite comparée à une autre borne obtenue par Jovanovski et Madras (2014).

Published online : 2016-01-21
@article{AFST_2015_6_24_4_935_0,
     author = {James P. Hobert and Kshitij Khare},
     title = {Computable upper bounds on the distance to stationarity for Jovanovski and Madras's Gibbs sampler},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {4},
     year = {2015},
     pages = {935-947},
     language = {en},
     url={afst.centre-mersenne.org/item/AFST_2015_6_24_4_935_0/}
}
Hobert, James P.; Khare, Kshitij. Computable upper bounds on the distance to stationarity for Jovanovski and Madras’s Gibbs sampler. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, pp. 935-947. https://afst.centre-mersenne.org/item/AFST_2015_6_24_4_935_0/

[1] Batir (N.).— Inequalities for the gamma function. Archiv der Mathematik, 91, p. 554-563 (2008). | MR 2465874 | Zbl 1165.33001

[2] Diaconis (P.), Khare (K.) and Saloff-Coste (L.).— Gibbs sampling, exponential families and orthogonal polynomials (with discussion). Statistical Science, 23, p. 151-200 (2008). | MR 2446500

[3] Diaconis (P.), Khare K. and Saloff-Coste (L.).— Gibbs sampling, conjugate priors and coupling. Sankhya ¯: The Indian Journal of Statistics, 72-A, p. 136-169 (2009). | MR 2658168 | Zbl 1209.60042

[4] Dudley (R. M.).— Real Analysis and Probability. Wadsworth, Belmont, CA (1989). | MR 982264 | Zbl 0686.60001

[5] Jovanovski (O.) and Madras (N.).— Convergence rates for hierarchical Gibbs samplers. Tech. rep., York University. ArXiv:1402.4733v2 (2014).

[6] Madras (N.) and Sezer (D.).— Quantitative bounds for Markov chain convergence: Wasserstein and total variation distance. Bernoulli, 16, p. 882-908 (2010). | MR 2730652 | Zbl 1284.60143

[7] Meyn (S. P.) and Tweedie (R. L.).— Markov Chains and Stochastic Stability. Springer-Verlag, London (1993). | MR 1287609 | Zbl 0925.60001

[8] Román (J. C.), Hobert (J. P.) and Presnell (B.).— On reparametrization and the Gibbs sampler. Statistics and Probability Letters, 91, p. 110-116 (2014). | MR 3208124 | Zbl 1296.60197

[9] Steinsaltz (D.).— Locally contractive iterated function systems. The Annals of Probability, 27, p. 1952-1979 (1999). | MR 1742896 | Zbl 0974.37037