Abstract
We consider the problem of testing the identity of a reversible Markov chain against a reference from a single trajectory of observations. Employing the recently introduced notion of a lumping-congruent Markov embedding, we show that, at least in a mildly restricted setting, testing identity to a reversible chain reduces to testing to a symmetric chain over a larger state space and recover state-of-the-art sample complexity for the problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As is customary in the property testing literature, we respectively write \(\varTheta , \mathcal {O}\) and \(\varOmega \) for tight, upper and lower bounds, and the tilda notation suppresses lower-order logarithmic factors in any parameter.
- 2.
General contrast functions under consideration satisfy identity of indiscernibles and non-negativity (e.g. proper metrics induced from matrix norms), and need not satisfy symmetry or the triangle inequality (e.g. information divergence rate between Markov processes).
- 3.
We note that [6] also slightly loosen the requirement of having a matching stationary distributions to being close in the sense where \(\left\| \pi /\overline{\pi } - 1 \right\| _\infty < \varepsilon \).
- 4.
If we wish to test for the identity of multiple chains against the same reference, we only need to perform this step once.
References
Canonne, C.L., et al.: Topics and techniques in distribution testing: A biased but representative sample. Found. Trends Commun. Inf. Theor. 19(6), 1032–1198 (2022)
Chan, S.O., Ding, Q., Li, S.H.: Learning and testing irreducible Markov chains via the \(k\)-cover time. In: Algorithmic Learning Theory. pp. 458–480. PMLR (2021)
Cherapanamjeri, Y., Bartlett, P.L.: Testing symmetric Markov chains without hitting. In: Proceedings of the Thirty-Second Conference on Learning Theory. Proceedings of Machine Learning Research, vol. 99, pp. 758–785. PMLR (2019)
Daskalakis, C., Dikkala, N., Gravin, N.: Testing symmetric Markov chains from a single trajectory. In: Conference On Learning Theory. pp. 385–409. PMLR (2018)
Diakonikolas, I., Kane, D.M.: A new approach for testing properties of discrete distributions. In: 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS). pp. 685–694. IEEE (2016)
Fried, S., Wolfer, G.: Identity testing of reversible Markov chains. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 151, pp. 798–817. PMLR (2022)
Goldreich, O.: The uniform distribution is complete with respect to testing identity to a fixed distribution. In: Electron. Colloquium Comput. Complex. vol. 23, p. 15 (2016)
Hayashi, M., Watanabe, S.: Information geometry approach to parameter estimation in Markov chains. Ann. Stat. 44(4), 1495–1535 (2016)
Kazakos, D.: The Bhattacharyya distance and detection between Markov chains. IEEE Trans. Inf. Theor. 24(6), 747–754 (1978)
Kemeny, J.G., Snell, J.L.: Finite Markov chains: with a new appendix Generalization of a fundamental matrix. Springer (1983)
Nagaoka, H.: The exponential family of Markov chains and its information geometry. In: The proceedings of the Symposium on Information Theory and Its Applications. vol. 28(2), pp. 601–604 (2005)
Paninski, L.: A coincidence-based test for uniformity given very sparsely sampled discrete data. IEEE Trans. Inf. Theor. 54(10), 4750–4755 (2008)
Rached, Z., Alajaji, F., Campbell, L.L.: Rényi’s divergence and entropy rates for finite alphabet Markov sources. IEEE Trans. Inf. Theor. 47(4), 1553–1561 (2001)
Valiant, G., Valiant, P.: An automatic inequality prover and instance optimal identity testing. SIAM J. Comput. 46(1), 429–455 (2017)
Čencov, N.N.: Algebraic foundation of mathematical statistics. Series Stat. 9(2), 267–276 (1978)
Čencov, N.N.: Statistical decision rules and optimal inference, transl. math. monographs, vol. 53. Amer. Math. Soc., Providence-RI (1981)
Waggoner, B.: \(l_p\) testing and learning of discrete distributions. In: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science. pp. 347–356 (2015)
Wolfer, G., Kontorovich, A.: Minimax testing of identity to a reference ergodic Markov chain. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. vol. 108, pp. 191–201. PMLR (2020)
Wolfer, G., Watanabe, S.: Information geometry of reversible Markov chains. Inf. Geom. 4(2), 393–433 (12 2021)
Wolfer, G., Watanabe, S.: Geometric aspects of data-processing of Markov chains (2022), arXiv:2203.04575
Acknowledgements
GW is supported by the Special Postdoctoral Researcher Program (SPDR) of RIKEN and by the Japan Society for the Promotion of Science KAKENHI under Grant 23K13024. SW is supported in part by the Japan Society for the Promotion of Science KAKENHI under Grant 20H02144.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Wolfer, G., Watanabe, S. (2023). Geometric Reduction for Identity Testing of Reversible Markov Chains. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_32
Download citation
DOI: https://doi.org/10.1007/978-3-031-38271-0_32
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-38270-3
Online ISBN: 978-3-031-38271-0
eBook Packages: Computer ScienceComputer Science (R0)