Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 1538-1565.

Predictive characterization of mixtures of Markov chains

Abstract

Predictive constructions are a powerful way of characterizing the probability laws of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps in designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process $\mathbf{X}$. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.

Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 1538-1565.

Dates
Revised: April 2015
First available in Project Euclid: 17 March 2017

https://projecteuclid.org/euclid.bj/1489737617

Digital Object Identifier
doi:10.3150/15-BEJ787

Mathematical Reviews number (MathSciNet)
MR3624870

Zentralblatt MATH identifier
06714311

Citation

Fortini, Sandra; Petrone, Sonia. Predictive characterization of mixtures of Markov chains. Bernoulli 23 (2017), no. 3, 1538--1565. doi:10.3150/15-BEJ787. https://projecteuclid.org/euclid.bj/1489737617

References

• [1] Aldous, D.J. (1985). Exchangeability and related topics. In École D’été de Probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math. 1117 1–198. Berlin: Springer.
• [2] Bacallado, S. (2011). Bayesian analysis of variable-order, reversible Markov chains. Ann. Statist. 39 838–864.
• [3] Bacallado, S., Favaro, S. and Trippa, L. (2013). Bayesian nonparametric analysis of reversible Markov chains. Ann. Statist. 41 870–896.
• [4] Beal, M.J., Ghahramani, Z. and Rasmussen, C.E. (2002). The infinite hidden Markov model. In Adv. Neural Inf. Process. Syst. 14 577–584. Cambridge, MA: MIT Press.
• [5] Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
• [6] Coppersmith, D. and Diaconis, P. (1987). Random walk with reinforcement. Unpublished manuscript.
• [7] de Finetti, B. (1937). La prévision : Ses lois logiques, ses sources subjectives. Ann. Inst. Henri Poincaré 7 1–68.
• [8] de Finetti, B. (1959). La probabilità e la statistica nei rapporti con l’induzione secondo i diversi punti di vista. In Centro Internazionale Matematico Estivo (CIME), Induzione e Statistica 1–115. Roma: Cremonese. English translation in B. de Finetti (1972). Probability, Induction and Statistics, 147–227.
• [9] Diaconis, P. (1988). Recent progress on de Finetti’s notions of exchangeability. In Bayesian Statistics, 3 (Valencia, 1987) (J.M. Bernardo, M.H. DeGroot, D.V. Lindley and A.F.M. Smith, eds.). Oxford Sci. Publ. 111–125. New York: Oxford Univ. Press.
• [10] Diaconis, P. and Freedman, D. (1980). de Finetti’s theorem for Markov chains. Ann. Probab. 8 115–130.
• [11] Diaconis, P. and Rolles, S.W.W. (2006). Bayesian analysis for reversible Markov chains. Ann. Statist. 34 1270–1292.
• [12] Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7 269–281.
• [13] Ewens, W.J. (1972). The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3 87–112.
• [14] Fortini, S., Ladelli, L., Petris, G. and Regazzini, E. (2002). On mixtures of distributions of Markov chains. Stochastic Process. Appl. 100 147–165.
• [15] Fortini, S., Ladelli, L. and Regazzini, E. (2000). Exchangeability, predictive distributions and parametric models. Sankhya, Ser. A 62 86–109.
• [16] Fortini, S. and Petrone, S. (2012). Hierarchical reinforced urn processes. Statist. Probab. Lett. 82 1521–1529.
• [17] Griffiths, T.L. and Ghahramani, Z. (2006). Infinite latent feature models and the Indian buffet process. In Adv. Neural Inf. Process. Syst. 18 475–482. Cambridge, MA: MIT Press.
• [18] Hoppe, F.M. (1984). Pólya-like urns and the Ewens’ sampling formula. J. Math. Biol. 20 91–94.
• [19] Keane, M.S. and Rolles, S.W.W. (2000). Edge-reinforced random walk on finite graphs. In Infinite Dimensional Stochastic Analysis (Amsterdam, 1999) (P. Clement, F. den Hollander, J. van Neerven and B. de Pagter, eds.). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. 52 217–234. Amsterdam: R. Neth. Acad. Arts Sci.
• [20] Merkl, F. and Rolles, S.W.W. (2005). Edge-reinforced random walk on a ladder. Ann. Probab. 33 2051–2093.
• [21] Merkl, F. and Rolles, S.W.W. (2007). A random environment for linearly edge-reinforced random walks on infinite graphs. Probab. Theory Related Fields 138 157–176.
• [22] Merkl, F. and Rolles, S.W.W. (2007). Asymptotic behavior of edge-reinforced random walks. Ann. Probab. 35 115–140.
• [23] Merkl, F. and Rolles, S.W.W. (2009). Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 1679–1714.
• [24] Muliere, P., Secchi, P. and Walker, S.G. (2000). Urn schemes and reinforced random walks. Stochastic Process. Appl. 88 59–78.
• [25] Orbanz, P. (2009). Functional conjugacy in parametric Bayesian models. Technical report, Univ. Cambridge.
• [26] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229–1241.
• [27] Pemantle, R. (2007). A survey of random processes with reinforcement. Probab. Surv. 4 1–79.
• [28] Pemantle, R.A. (1988). Random processes with reinforcement. Ph.D. thesis, Dept. Mathematics, Massachusetts Institute of Technology.
• [29] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21–39.
• [30] Pitman, J. (1996). Some developments of the Blackwell–MacQueen urn scheme. In Statistics, Probability and Game Theory. Institute of Mathematical Statistics Lecture Notes – Monograph Series 30 245–267. Hayward, CA: IMS.
• [31] Rolles, S.W.W. (2003). How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 243–260.
• [32] Rolles, S.W.W. (2006). On the recurrence of edge-reinforced random walk on ${\mathbb{Z}}\times G$. Probab. Theory Related Fields 135 216–264.
• [33] Sabot, C. and Tarrès, P. (2015). Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. (JEMS) 17 2353–2378.
• [34] Teh, Y.W. and Jordan, M.I. (2010). Hierarchical Bayesian nonparametric models with applications. In Bayesian Nonparametrics (N.L. Hjort, C. Holmes, P. Muller and S.G. Walker, eds.). Camb. Ser. Stat. Probab. Math. 158–207. Cambridge: Cambridge Univ. Press.
• [35] Teh, Y.W., Jordan, M.I., Beal, M.J. and Blei, D.M. (2006). Hierarchical Dirichlet processes. J. Amer. Statist. Assoc. 101 1566–1581.
• [36] Thibaux, R. and Jordan, M.I. (2007). Hierarchical beta processes and the Indian buffet process. In JMLR Workshops and Proceedings: AISTATS 2007 2 564–571.
• [37] Wade, S., Mongelluzzo, S. and Petrone, S. (2011). An enriched conjugate prior for Bayesian nonparametric inference. Bayesian Anal. 6 359–385.
• [38] Walker, S. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25 1762–1780.
• [39] Walker, S. and Muliere, P. (1999). A characterization of a neutral to the right prior via an extension of Johnson’s sufficientness postulate. Ann. Statist. 27 589–599.
• [40] Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge Univ. Press.
• [41] Zabell, S.L. (1982). W. E. Johnson’s “sufficientness” postulate. Ann. Statist. 10 1090–1099.
• [42] Zabell, S.L. (1995). Characterizing Markov exchangeable sequences. J. Theoret. Probab. 8 175–178.
• [43] Zaman, A. (1984). Urn models for Markov exchangeability. Ann. Probab. 12 223–229.