Bayesian Analysis

On a Nonparametric Change Point Detection Model in Markovian Regimes

Asael Fabian Martínez and Ramsés H. Mena

Full-text: Open access


Change point detection models aim to determine the most probable grouping for a given sample indexed on an ordered set. For this purpose, we propose a methodology based on exchangeable partition probability functions, specifically on Pitman’s sampling formula. Emphasis will be given to the Markovian case, in particular for discretely observed Ornstein-Uhlenbeck diffusion processes. Some properties of the resulting model are explained and posterior results are obtained via a novel Markov chain Monte Carlo algorithm.

Article information

Bayesian Anal., Volume 9, Number 4 (2014), 823-858.

First available in Project Euclid: 21 November 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian nonparametric Change point detection Ornstein-Uhlenbeck process Two-parameter Poisson-Dirichlet process


Martínez, Asael Fabian; Mena, Ramsés H. On a Nonparametric Change Point Detection Model in Markovian Regimes. Bayesian Anal. 9 (2014), no. 4, 823--858. doi:10.1214/14-BA878.

Export citation


  • Aggarwal, R., Inclán, C., and Leal, R. (1999). “Volatility in emerging stock markets.” Journal of Financial and Quantitative Analysis, 34: 1–17.
  • Allen, D. E., McAleer, M., Powell, R. J., and Kumar-Singh, A. (2013). “Nonparametric Multiple Change Point Analysis of the Global Financial Crisis.” URL
  • Barry, D. and Hartigan, J. (1992). “Product partition models for change point problems.” The Annals of Statistics, 20(1): 260–279.
  • Barry, D. and Hartigan, J. A. (1993). “A Bayesian Analysis for Change Point Problems.” Journal of the American Statistical Association, 88(421): 309–319.
  • Benavides, G. and Capistrán, C. (2009). “Una Nota sobre la Volatilidad de la Tasa de Interés y del Tipo de Cambio bajo Diferentes Instrumentos de Política Monetaria: México 1998–2008.” Working Paper 2009-10, Banco de México, México.
  • Bhattacharya, R. N., Gupta, V. K., and Waymire, E. (1983). “The Hurst effect under trends.” Journal of Applied Probability, 20: 649–662.
  • Braun, J. V., Braun, R. K., and Müller, H. G. (2000). “Multiple Changepoint Fitting via Quasilikelihood, with Application to DNA Sequence Segmentation.” Biometrika, 87(2): 301–314.
  • Brodsky, B. E. and Darkhovsky, B. S. (1993). Nonparametric Methods in Change–Point Problems. Springer.
  • Bubula, A. and Ötker-Robe, I. (2002). “The Evolution of Exchange Rate Regimes Since 1990: Evidence from De Facto Policies.” Working Paper 02/155, International Monetary Fund.
  • Chen, J. and Gupta, A. K. (1997). “Testing and locating variance changepoints with application to stock prices.” Journal of the American Statistical Association, 92(438): 739–747.
  • — (2011). Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance. Birkhauser, 2nd edition.
  • Chernoff, H. and Zacks, S. (1964). “Estimating the Current Mean of a Normal Distribution which is Subjected to Changes in Time.” Annals of Mathematical Statistics, 35(3): 999–1018.
  • Dahl, D. B. (2006). “Model-Based Clustering for Expression Data via a Dirichlet Process Mixture Model.” In Do, K.-A., Müller, P., and Vannucci, M. (eds.), Bayesian Inference for Gene Expression and Proteomics, 201–218. Cambridge University Press.
  • De Blasi, P., Favaro, S., Lijoi, A., Mena, R., Prünster, I., and Ruggiero, M. (2013). “Are Gibbs-Type Priors the Most Natural Generalization of the Dirichlet Process?” IEEE Transactions on Pattern Analysis and Machine Intelligence. URL
  • de Finetti, B. (1931). Funzione Caratteristica Di un Fenomeno Aleatorio, 251–299. 6. Memorie. Academia Nazionale del Linceo.
  • Dobigeon, N. and Toumeret, J. Y. (2007). “Joint segmentation of wind speed and direction using a hierarchical model.” Computational Statistics and Data Analysis, 51: 5603–5621.
  • Fearnhead, P. and Liu, Z. (2007). “Online Inference for Multiple Changepoint Problems.” Journal of the Royal Statistical Society, Series B (Statistical Methodology), 69: 589–605.
  • Ferguson, T. (1973). “A Bayesian analysis of some nonparametric problems.” The Annals of Statistics, 1(2): 209–230.
  • Fuentes-García, R., Mena, R. H., and Walker, S. G. (2010). “A Probability for Classification Based on the Dirichlet Process Mixture Model.” Journal of Classification, 27: 389–403.
  • Gelman, A. and Rubin, D. B. (1992). “Inference from Iterative Simulation Using Multiple Sequences.” Statistical Science, 7(4).
  • Gilks, W. R., Best, N. G., and Tan, K. K. C. (1995). “Adaptive rejection Metropolis sampling.” Applied Statistics, 44: 455–472.
  • Godsill, S. (2001). “On the Relationship Between Markov Chain Monte Carlo Methods for Model Uncertainty.” Journal of Computational and Graphical Statistics, 10(2): 230–248.
  • Green, P. J. (1995). “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination.” Biometrika, 82(4): 711–732.
  • Harding, D. and Pagan, A. R. (2008). “Business cycle measurement.” In Durlauf, S. N. and Blume, L. E. (eds.), The New Palgrave Dictionary of Economics. Palgrave Macmillan, 2nd edition.
  • Jara, A., Lesaffre, E., De Iorio, M., and Quintana, F. (2010). “Bayesian semiparametric inference for multivariate doubly–interval–censored data.” Annals of Applied Statistics, 4(4): 2126–2149.
  • Jochmann, M. (2010). “Modeling U.S. Inflation Dynamics: A Bayesian Nonparametric Approach.” Technical Report 2010–06, Scottish Institute for Research in Economics (SIRE).
  • Kander, Z. and Zacks, S. (1966). “Test Procedures for Possible Changes in Parameters of Statistical Distributions Occurring at Unknown Time Points.” Annals of Mathematical Statistics, 37(5): 1196–1210.
  • Kaplan, A. Y. and Shishkin, S. L. (2000). “Application of the change-point analysis to the investigation of the brain’s electrical activity.” In Brodsky, B. E. and Darkhovsky, B. S. (eds.), Non-Parametric Statistical Diagnosis: Problems and Methods, 333–388. Springer.
  • Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer-Verlag.
  • Kingman, J. (1975). “Random discrete distributions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 37(1): 1–22.
  • Lavielle, M. and Teyssière, G. (2007). “Adaptive Detection of Multiple Change-Points in Asset Price Volatility.” In Long Memory in Economics, 129–156. Springer Berlin Heidelberg.
  • Lijoi, A., Mena, R. H., and Prünster, I. (2007a). “Bayesian nonparametric estimation of the probability of discovering a new species.” Biometrika, 94: 769–786.
  • — (2007b). “Controlling the reinforcement in Bayesian non-parametric mixture models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69(4): 715–740.
  • Lijoi, A. and Prünster, I. (2010). “Models beyond the Dirichlet process.” In Hjort, N. L., Holmes, C. C., Müller, P., and Walker, S. G. (eds.), Bayesian Nonparametrics, 80–136. Cambridge University Press.
  • Loschi, R. H. and Cruz, F. R. B. (2005). “Extension to the product partition model: computing the probability of a change.” Computational Statistics & Data Analysis, 48(2): 255–268.
  • Loschi, R. H., Cruz, F. R. B., Iglesias, P. L., and Arellano-Valle, R. B. (2003). “A Gibbs sampling scheme to the product partition model: an application to change point problems.” Computers & Operations Research, 30(3): 463–482.
  • Mikosch, T. and Stărică, C. (2004). “Changes of structure in financial time series and the GARCH model.” REVSTAT - Statistical Journal, 2(1): 41–73.
  • Minin, V. N., Dorman, K. S., Fang, F., and Suchard, M. A. (2007). “Phylogenetic Mapping of Recombination Hotspots in Human Immunodeficiency Virus via Spatially Smoothed Change-Point Processes.” Genetics, 175(4): 1773–1785.
  • Mira, A. and Petrone, S. (1996). “Bayesian hierarchical nonparametric inference for change–point problems.” In Bernardo, J., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics 5, 693–703. Oxford University Press.
  • Monteiro, J. V. D., Assunção, R., and Loschi, R. H. (2011). “Product partition models with correlated parameters.” Bayesian Analysis, 6(4): 691–726.
  • Muliere, P. and Scarsini, M. (1985). “Change–point problems: A Bayesian nonparametric approach.” Aplikace matematiky, 30(6): 397–402.
  • Müller, P., Quintana, F., and Rosner, G. L. (2011). “A Product Partition Model With Regression on Covariates.” Journal of Computational and Graphical Statistics, 20(1): 260–278.
  • Nieto-Barajas, L. E. and Contreras-Cristan, A. (2014). “A Bayesian Nonparametric Approach for Time Series Clustering.” Bayesian Analysis, 9(1): 147–170.
  • Park, J.-H. and Dunson, D. B. (2010). “Bayesian generalized product partition model.” Statistica Sinica, 20(20): 1203–1226.
  • Perman, M., Pitman, J., and Yor, M. (1992). “Size–biased sampling of Poisson point processes and excursions.” Probability Theory and Related Fields, 92: 21–39.
  • Perreault, L., Bernier, J., Bobée, B., and Parent, E. (2000). “Bayesian change-point analysis in hydrometeorological time series. Part 1. The normal model revisited.” Journal of Hydrology, 235(3–4): 221–241.
  • Pitman, J. (2006). Combinatorial stochastic processes. Ecole d’eté de probabilités de Saint-Flour XXXII - 2002. Springer.
  • Punskaya, E., Andrieu, C., Doucet, A., and Fitzgerald, W. J. (2002). “Bayesian curve fitting using MCMC with applications to signal segmentation.” IEEE Transactions on Signal Processing, 50: 747–758.
  • Quintana, F. A. and Iglesias, P. L. (2003). “Bayesian Clustering and Product Partition Models.” Journal of the Royal Statistical Society, Series B (Statistical Methodology), 65(2): 557–574.
  • Raftery, A. E. and Lewis, S. M. (1992). “How many iterations in the Gibbs sampler?” In Bernardo, J., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics 4, 763–773. Oxford University Press.
  • Regazzini, E., Lijoi, A., and Prünster, I. (2003). “Distributional results for means of normalized random measures with independent increments.” Annals of Statistics, 31: 560–585.
  • Reinhart, C. M. and Rogoff, K. S. (2004). “The Modern History of Exchange Rate Arrangements: A Reinterpretation.” The Quarterly Journal of Economics, 119(1): 1–48.
  • Smith, A. F. M. (1975). “A Bayesian approach to inference about a change–point in a sequence of random variables.” Biometrika, 62(2): 407–416.
  • Uhlenbeck, G. E. and Ornstein, L. (1930). “On the theory of Brownian motion.” Physical Review, 36(36): 823–841.
  • West, M. (1992). “Hyperparameter estimation in Dirichlet process mixture models.” Technical report, Institute of Statistics and Decision Sciences, Duke University.
  • Yao, Y.-C. (1984). “Estimation of a Noisy Discrete-Time Step Function: Bayes and Empirical Bayes Approaches.” Annals of Statistics, 12(4): 1434–1447.
  • Zantedeschi, D., Damien, P., and Polson, N. G. (2011). “Predictive Macro-Finance With Dynamic Partition Models.” Journal of the American Statistical Association, 106(494): 427–439.