IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v73y2000i2p199-220.html
   My bibliography  Save this article

The Construction of Multivariate Distributions from Markov Random Fields

Author

Listed:
  • Kaiser, Mark S.
  • Cressie, Noel

Abstract

We address the problem of constructing and identifying a valid joint probability density function from a set of specified conditional densities. The approach taken is based on the development of relations between the joint and the conditional densities using Markov random fields (MRFs). We give a necessary and sufficient condition on the support sets of the random variables to allow these relations to be developed. This condition, which we call the Markov random field support condition, supercedes a common assumption known generally as the positivity condition. We show how these relations may be used in reverse order to construct a valid model from specification of conditional densities alone. The constructive process and the role of conditions needed for its application are illustrated with several examples, including MRFs with multiway dependence and a spatial beta process.

Suggested Citation

  • Kaiser, Mark S. & Cressie, Noel, 2000. "The Construction of Multivariate Distributions from Markov Random Fields," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 199-220, May.
    • Handle: RePEc:eee:jmvana:v:73:y:2000:i:2:p:199-220
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(99)91878-X
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Li, Haijun & Scarsini, Marco & Shaked, Moshe, 1996. "Linkages: A Tool for the Construction of Multivariate Distributions with Given Nonoverlapping Multivariate Marginals," Journal of Multivariate Analysis, Elsevier, vol. 56(1), pages 20-41, January.
    2. Cramer, Erhard, 1998. "Conditional Iterative Proportional Fitting for Gaussian Distributions," Journal of Multivariate Analysis, Elsevier, vol. 65(2), pages 261-276, May.
    3. Cuadras, C. M., 1992. "Probability distributions with given multivariate marginals and given dependence structure," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 51-66, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Berti, Patrizia & Dreassi, Emanuela & Rigo, Pietro, 2014. "Compatibility results for conditional distributions," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 190-203.
    2. Dreassi, Emanuela & Rigo, Pietro, 2017. "A note on compatibility of conditional autoregressive models," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 9-16.
    3. Emily Casleton & Daniel J. Nordman & Mark S. Kaiser, 2022. "Modeling Transitivity in Local Structure Graph Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 389-417, June.
    4. Nail Kashaev & Natalia Lazzati, 2019. "Peer Effects in Random Consideration Sets," Papers 1904.06742, arXiv.org, revised May 2021.
    5. Kopciuszewski, Pawel, 2004. "An extension of the factorization theorem to the non-positive case," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 118-130, January.
    6. Christopher K. Wikle, 2003. "Hierarchical Models in Environmental Science," International Statistical Review, International Statistical Institute, vol. 71(2), pages 181-199, August.
    7. R. Reeves, 2004. "Efficient recursions for general factorisable models," Biometrika, Biometrika Trust, vol. 91(3), pages 751-757, September.
    8. Linda Khachatryan & Boris S. Nahapetian, 2023. "On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1743-1761, September.
    9. Mark S. Kaiser & Petruţa C. Caragea, 2009. "Exploring Dependence with Data on Spatial Lattices," Biometrics, The International Biometric Society, vol. 65(3), pages 857-865, September.
    10. Wang, Yuchung J. & Kuo, Kun-Lin, 2010. "Compatibility of discrete conditional distributions with structural zeros," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 191-199, January.
    11. Lee, Jaehyung & Kaiser, Mark S. & Cressie, Noel, 2001. "Multiway Dependence in Exponential Family Conditional Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 171-190, November.
    12. Noel Cressie & Craig Liu, 2001. "Binary Markov Mesh Models and Symmetric Markov Random Fields: Some Results on their Equivalence," Methodology and Computing in Applied Probability, Springer, vol. 3(1), pages 5-34, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nabil Kazi-Tani & Didier Rullière, 2019. "On a construction of multivariate distributions given some multidimensional marginals," Post-Print hal-01575169, HAL.
    2. Nabil Kazi-Tani & Didier Rullière, 2017. "On a construction of multivariate distributions given some multidimensional marginals," Working Papers hal-01575169, HAL.
    3. Sancetta, A., 2005. "Copula Based Monte Carlo Integration in Financial Problems," Cambridge Working Papers in Economics 0506, Faculty of Economics, University of Cambridge.
    4. Gautier Marti & Frank Nielsen & Philippe Donnat & S'ebastien Andler, 2016. "On clustering financial time series: a need for distances between dependent random variables," Papers 1603.07822, arXiv.org.
    5. Fernández-Ponce, J.M. & Pellerey, F. & Rodríguez-Griñolo, M.R., 2011. "A characterization of the multivariate excess wealth ordering," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 410-417.
    6. Bouye, Eric & Durlleman, Valdo & Nikeghbali, Ashkan & Riboulet, Gaël & Roncalli, Thierry, 2000. "Copulas for finance," MPRA Paper 37359, University Library of Munich, Germany.
    7. Belzunce, Félix & Ruiz, José M. & Suárez-Llorens, Alfonso, 2008. "On multivariate dispersion orderings based on the standard construction," Statistics & Probability Letters, Elsevier, vol. 78(3), pages 271-281, February.
    8. Alfred Müller & Marco Scarsini, 2001. "Stochastic Comparison of Random Vectors with a Common Copula," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 723-740, November.
    9. Cuadras, C. M. & Atkinson, R. A. & Fortiana, J., 1997. "Probability densities from distances and discrimination," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 405-411, May.
    10. Gramer Erhard, 2000. "Probability Measures With Given Marginals And Conditionals: I-Projections And Conditional Iterative Proportional Fitting," Statistics & Risk Modeling, De Gruyter, vol. 18(3), pages 311-330, March.
    11. Sancetta, A. & Nikanrova, A., 2005. "Forecasting and Prequential Validation for Time Varying Meta-Elliptical Distributions with a Study of Commodity Futures Prices," Cambridge Working Papers in Economics 0516, Faculty of Economics, University of Cambridge.
    12. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    13. Sugata Ghosh & Subhajit Dutta & Marc G. Genton, 2017. "A note on inconsistent families of discrete multivariate distributions," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-13, December.
    14. Cuadras, C. M., 2002. "On the Covariance between Functions," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 19-27, April.
    15. Li, Haijun & Scarsini, Marco & Shaked, Moshe, 1999. "Dynamic Linkages for Multivariate Distributions with Given Nonoverlapping Multivariate Marginals," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 54-77, January.
    16. Xuan Vinh Doan & Karthik Natarajan, 2012. "On the Complexity of Nonoverlapping Multivariate Marginal Bounds for Probabilistic Combinatorial Optimization Problems," Operations Research, INFORMS, vol. 60(1), pages 138-149, February.
    17. Fan, Yanqin & Henry, Marc, 2023. "Vector copulas," Journal of Econometrics, Elsevier, vol. 234(1), pages 128-150.
    18. Bairamov, Ismihan & Khaledi, Baha-Eldin & Shaked, Moshe, 2014. "Stochastic comparisons of order statistics and their concomitants," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 105-115.
    19. Marian Núñez & Angel Villarroya & José María Oller, 2003. "Minimum Distance Probability Discriminant Analysis for Mixed Variables," Biometrics, The International Biometric Society, vol. 59(2), pages 248-253, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:73:y:2000:i:2:p:199-220. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.