Statistics and Computing

, Volume 22, Issue 4, pp 981–993 | Cite as

Implied distributions in multiple change point problems

  • J. A. D. Aston
  • J. Y. Peng
  • D. E. K. Martin


A method for efficiently calculating exact marginal, conditional and joint distributions for change points defined by general finite state Hidden Markov Models is proposed. The distributions are not subject to any approximation or sampling error once parameters of the model have been estimated. It is shown that, in contrast to sampling methods, very little computation is needed. The method provides probabilities associated with change points within an interval, as well as at specific points.


Finite Markov chain imbedding Hidden Markov models Change point probability Run length distributions Generalised change points Waiting time distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

11222_2011_9268_MOESM1_ESM.pdf (273 kb)
Implied distributions in multiplechange point problems—supplementary information (PDF 272 KB)


  1. Albert, J.H., Chib, S.: Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts. J. Bus. Econ. Stat. 11, 1–15 (1993) CrossRefGoogle Scholar
  2. Aston, J.A.D., Martin, D.E.K.: Distributions associated with general runs and patterns in hidden Markov models. Ann. Appl. Stat. 1, 585–611 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Balakrishnan, N., Koutras, M.: Runs and Scans with Applications. Wiley, New York (2002) zbMATHGoogle Scholar
  4. Barry, D., Hartigan, J.A.: Product partition models for change point problems. Ann. Stat. 20, 260–279 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  5. Barry, D., Hartigan, J.A.: A Bayesian analysis for change-point problems. J. Am. Stat. Assoc. 88, 309–319 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  6. Billio, M., Monfort, A.: Switching state space models: likelihood functions, filtering and smoothing. J. Stat. Plan. Inference 68, 65–103 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  7. Cappé, O., Moulines, E., Rydén, T.: Inference in Hidden Markov Models. Springer, Berlin (2005) zbMATHGoogle Scholar
  8. Carpenter, J., Clifford, P., Fearnhead, P.: An improved particle filter for non-linear problems. IEE Proc. Radar Sonar Navig. 146, 2–7 (1999) CrossRefGoogle Scholar
  9. Chib, S.: Estimation and comparison of multiple change-point models. J. Econom. 86, 221–241 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  10. Durbin, R., Eddy, S., Krogh, A., Mitchison, G.: Biological Sequence Analysis. Cambridge University Press, Cambridge (1998) zbMATHCrossRefGoogle Scholar
  11. Eddy, S.R.: What is a hidden Markov model? Nat. Biotechnol. 22, 1315–1316 (2004) CrossRefGoogle Scholar
  12. Fearnhead, P.: Exact and efficient Bayesian inference for multiple changepoint problems. Stat. Comput. 16, 203–213 (2006) MathSciNetCrossRefGoogle Scholar
  13. Fearnhead, P., Liu, Z.: Online inference for multiple changepoint problems. J. R. Stat. Soc., Ser. B 69, 589–605 (2007) MathSciNetCrossRefGoogle Scholar
  14. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968) zbMATHGoogle Scholar
  15. Frühwirth-Schnatter, S.: Finite Mixture and Markov Switching Models. Springer, Berlin (2006) zbMATHGoogle Scholar
  16. Fu, J.C., Koutras, M.V.: Distribution theory of runs: a Markov chain approach. J. Am. Stat. Assoc. 89, 1050–1058 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  17. Guédon, Y.: Exploring the state sequence space for hidden Markov and semi-Markov chains. Comput. Stat. Data Anal. 51, 2379–2409 (2007) zbMATHCrossRefGoogle Scholar
  18. Guédon, Y.: Exploring the segmentation space for the assessment of multiple change-point models. Technical Report 6619, INRIA (2008) Google Scholar
  19. Hamilton, J.D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Juang, B.H., Rabiner, L.R.: Hidden Markov models for speech recognition. Technometrics 33, 251–272 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kim, C.-J.: Dynamic linear models with Markov-switching. J. Econom. 60, 1–22 (1994) zbMATHCrossRefGoogle Scholar
  22. Kschischang, F.R., Frey, B.J., Loeliger, H.-A.: Factor graphs and the sum-product algorithm. IEEE Trans. Inf. Theory 47, 498–519 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  23. MacDonald, I.L., Zucchini, W.: Hidden Markov and Other Models for Discrete-valued Time Series. Chapman & Hall, London (1997) zbMATHGoogle Scholar
  24. Martin, A.D., Quinn, K.M., Park, J.H.: MCMCpack: Markov chain Monte Carlo in R. J. Stat. Softw. 42(9), 1–22 (2011) Google Scholar
  25. R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2011). ISBN 3-900051-07-0 Google Scholar
  26. Rabiner, L.R.: A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77, 257–286 (1989) CrossRefGoogle Scholar
  27. Sims, C.A., Zha, T.: Were there regime switches in US monetary policy? Am. Econ. Rev. 96, 54–81 (2006) CrossRefGoogle Scholar
  28. Viterbi, A.: Error bounds for convolution codes and an asymptotically optimal decoding algorithm. IEEE Trans. Inf. Theory 13, 260–269 (1967) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • J. A. D. Aston
    • 1
  • J. Y. Peng
    • 2
    • 3
  • D. E. K. Martin
    • 4
  1. 1.Centre for Research in Statistical MethodologyUniversity of WarwickCoventryUK
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiTaiwan
  3. 3.Institute of Biomedical InformaticsNational Yang-Ming UniversityTaipeiTaiwan
  4. 4.Dept. of StatisticsNorth Carolina State UniversityRaleighUSA

Personalised recommendations