Skip to main content

Exact posterior distributions and model selection criteria for multiple change-point detection problems

Abstract

In segmentation problems, inference on change-point position and model selection are two difficult issues due to the discrete nature of change-points. In a Bayesian context, we derive exact, explicit and tractable formulae for the posterior distribution of variables such as the number of change-points or their positions. We also demonstrate that several classical Bayesian model selection criteria can be computed exactly. All these results are based on an efficient strategy to explore the whole segmentation space, which is very large. We illustrate our methodology on both simulated data and a comparative genomic hybridization profile.

This is a preview of subscription content, access via your institution.

References

  • Akaike, H.: Information theory as an extension of the maximum likelihood principle. In: Petrov, B., Csaki, F. (eds.) Second International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest (1973)

    Google Scholar 

  • Bai, J., Perron, P.: Computation and analysis of multiple structural change models. J. Appl. Econ. 18, 1–22 (2003)

    Article  Google Scholar 

  • Baraud, Y., Giraud, C., Huet, S.: Gaussian model selection with unknown variance. Ann. Stat. 37(2), 630–672 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  • Biernacki, C., Celeux, G., Govaert, G.: Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intell. 22(7), 719–725 (2000)

    Article  Google Scholar 

  • Biernacki, C., Celeux, G., Govaert, G.: Exact and Monte-Carlo calculation of integrated likelihoods for the latent class model. J. Stat. Plan. Inference 140, 2191–3002 (2010)

    MathSciNet  Article  Google Scholar 

  • Birgé, L., Massart, P.: Minimal penalties for Gaussian model selection. Probab. Theory Relat. Fields 138, 33–73 (2007)

    MATH  Article  Google Scholar 

  • Braun, R.-K., Braun, J.-V., Müller, H.-G.: Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika 87, 301–314 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  • Carlin, B.P., Chib, S.: Bayesian model choice via Markov chain Monte Carlo methods. J. R. Stat. Soc., Ser. B, Stat. Methodol. 57(3), 473–484 (1995). ArticleType: research-article/Full publication date: 1995/Copyright © 1995 Royal Statistical Society

    MATH  Google Scholar 

  • Chen, C., Chan, J., Gerlach, R., Hsieh, W.: A comparison of estimators for regression models with change points (2010). doi:10.1007/s11222-010-9177-0

  • Congdon, P.: Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Comput. Stat. Data Anal. 50(2), 346–357 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  • Congdon, P.: Model weights for model choice and averaging. Stat. Methodol. 4(2), 143–157 (2007)

    MathSciNet  Article  Google Scholar 

  • Feder, P.I.: The loglikelihood ratio in segmented regression. Ann. Stat. 3(1), 84–97 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  • Godsill, S.J.: On the relationship between Markov chain Monte Carlo methods for model uncertainty. J. Comput. Graph. Stat. 10, 230–248 (2001)

    MathSciNet  Article  Google Scholar 

  • Guédon, Y.: Explorating the segmentation space for the assessment of multiple change-points models. Technical report, Preprint INRIA n°6619 (2008)

  • Husková, M., Kirch, C.: Bootstrapping confidence intervals for the change-point of time series. J. Time Ser. Anal. 29(6), 947–972 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  • Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Stat. Assoc. 90, 773–795 (1995)

    MATH  Article  Google Scholar 

  • Lavielle, M.: Using penalized contrasts for the change-point problem. Signal Process. 85(8), 1501–1510 (2005)

    MATH  Article  Google Scholar 

  • Lebarbier, E.: Detecting multiple change-points in the mean of Gaussian process by model selection. Signal Process. 85, 717–736 (2005)

    MATH  Article  Google Scholar 

  • Lebarbier, E., Mary-Huard, T.: Une introduction au critère BIC : fondements théoriques et interprétation. J. Soc. Fr. Stat. 147(1), 39–57 (2006)

    MathSciNet  Google Scholar 

  • Lee, C.-B.: Estimating the number of change points in a sequence of independent normal random variables. Stat. Probab. Lett. 25(3), 241–8 (1995)

    MATH  Article  Google Scholar 

  • Muggeo, V.M.: Estimating regression models with unknown break-points. Stat. Med. 22(19), 3055–3071 (2003)

    Article  Google Scholar 

  • Picard, F., Robin, S., Lavielle, M., Vaisse, C., Daudin, J.-J.: A statistical approach for array CGH data analysis. BMC Bioinform. 6(27), 1 (2005). www.biomedcentral.com/1471-2105/6/27

    Google Scholar 

  • Pinkel, D., Segraves, R., Sudar, D., Clark, S., Poole, I., Kowbel, D., Collins, C., Kuo, W., Chen, C., Zhai, Y., Dairkee, S., Ljung, B., Gray, J.: High resolution analysis of DNA copy number variation using comparative genomic hybridization to microarrays. Nat. Genet. 20, 207–211 (1998)

    Article  Google Scholar 

  • Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)

    MATH  Article  Google Scholar 

  • Scott, S.L.: Bayesian methods for hidden Markov models: Recursive computing in the 21st century. J. Am. Stat. Assoc. 97(457), 337–351 (2002). ArticleType: research-article/Full publication date: Mar., 2002/Copyright © 2002 American Statistical Association

    MATH  Article  Google Scholar 

  • Spiegelhalter, D., Best, N., Carlin, B., van der Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. B 64(4), 583–639 (2002)

    MATH  Article  Google Scholar 

  • Toms, J.D., Lesperance, M.L.: Piecewise regression: A tool for identifying ecological thresholds. Ecology 84(8), 2034–2041 (2003)

    Article  Google Scholar 

  • Yao, Y.-C.: Estimating the number of change-points via Schwarz’ criterion. Stat. Probab. Lett. 6(3), 181–189 (1988)

    MATH  Article  Google Scholar 

  • Zhang, N.R., Siegmund, D.O.: A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data. Biometrics 63(1), 22–32 (2007)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Rigaill.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rigaill, G., Lebarbier, E. & Robin, S. Exact posterior distributions and model selection criteria for multiple change-point detection problems. Stat Comput 22, 917–929 (2012). https://doi.org/10.1007/s11222-011-9258-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-011-9258-8

Keywords

  • Bayesian model selection
  • change-point detection
  • BIC
  • DIC
  • ICL
  • posterior distribution of change-points
  • posterior distribution of segments