Exponential inequalities for VLMC empirical trees
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 219-229.

A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function of the size of the sample. As a consequence we prove the almost sure consistency of the algorithm Context. We also derive exponential upper bounds for type I errors and for the probability of underestimation of the context tree. The constants appearing in the bounds are all explicit and obtained in a constructive way.

DOI : https://doi.org/10.1051/ps:2007035
Classification : 62M05,  60G99
Mots clés : variable length Markov chain, context tree, algorithm context, weak dependance
@article{PS_2008__12__219_0,
     author = {Galves, Antonio and Maume-Deschamps, V\'eronique and Schmitt, Bernard},
     title = {Exponential inequalities for {VLMC} empirical trees},
     journal = {ESAIM: Probability and Statistics},
     pages = {219--229},
     publisher = {EDP-Sciences},
     volume = {12},
     year = {2008},
     doi = {10.1051/ps:2007035},
     zbl = {1182.62165},
     mrnumber = {2374639},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007035/}
}
TY  - JOUR
AU  - Galves, Antonio
AU  - Maume-Deschamps, Véronique
AU  - Schmitt, Bernard
TI  - Exponential inequalities for VLMC empirical trees
JO  - ESAIM: Probability and Statistics
PY  - 2008
DA  - 2008///
SP  - 219
EP  - 229
VL  - 12
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007035/
UR  - https://zbmath.org/?q=an%3A1182.62165
UR  - https://www.ams.org/mathscinet-getitem?mr=2374639
UR  - https://doi.org/10.1051/ps:2007035
DO  - 10.1051/ps:2007035
LA  - en
ID  - PS_2008__12__219_0
ER  - 
Galves, Antonio; Maume-Deschamps, Véronique; Schmitt, Bernard. Exponential inequalities for VLMC empirical trees. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 219-229. doi : 10.1051/ps:2007035. http://www.numdam.org/articles/10.1051/ps:2007035/

[1] G. Bejerano and G. Yona, Variations on probabilistic suffix trees: statistical modeling and prediction of protein families. Bioinformatics 17 (2001) 23-43.

[2] P. Bühlmann and A. Wyner, Variable length Markov chains. Ann. Statist. 27 (1999) 480-513. | MR 1714720 | Zbl 0983.62048

[3] I. Csiszár, Large-scale typicality of Markov sample paths and consistency of MDL order estimators. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48 (2002) 1616-1628. | MR 1909476 | Zbl 1060.62092

[4] I. Csiszár and Z. Talata, Context tree estimation for not necessarily finite memory processes via BIC and MDL, manuscript (2005). | MR 2238067 | Zbl 1284.94027

[5] J. Dedecker and P. Doukhan, A new covariance inequality and applications. Stochastic Process. Appl. 106 (2003) 63-80. | MR 1983043 | Zbl 1075.60513

[6] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics. Prob. Theory Relat. Fields 132 (2005) 203-236. | MR 2199291 | Zbl 1061.62058

[7] P. Ferrari and A. Galves, Coupling and regeneration for stochastic processes2000).

[8] F. Ferrari and A. Wyner, Estimation of general stationary processes by variable length Markov chains. Scand. J. Statist. 30 (2003) 459-480. | MR 2002222 | Zbl 1034.62080

[9] F. Leonardi and A. Galves, Sequence Motif identification and protein classification using probabilistic trees. Lect. Notes Comput. Sci. 3594 (2005) 190-193.

[10] V. Maume-Deschamps, Exponential inequalities and estimation of conditional probabilities in Dependence in probability and statistics, Lect. Notes in Stat., Vol. 187, P. Bertail, P. Doukhan and P. Soulier Eds. Springer (2006). | MR 2283253

[11] J. Rissanen, A universal data compression system. IEEE Trans. Inform. Theory 29 (1983) 656-664. | MR 730903 | Zbl 0521.94010

[12] T.J. Tjalkens and F.M.J.F. Willems, Implementing the context-tree weighting method: arithmetic coding. Recent advances in interdisciplinary mathematics (Portland, ME, 1997). J. Combin. Inform. System Sci. 25 (2000) 49-58. | MR 1807734

[13] F.M. Willems, Y.M. Shtarkov and T.J Tjalkens, The context-tree weighting method: basic properties. IEEE Trans. Inform. Theory 41 (1995) 653-664. | Zbl 0837.94011

Cité par Sources :