Skip to main content
SpringerLink
Log in

Graphical Latent Structure Testing

  • Chapter
  • First Online:
Advances in Latent Variables

Part of the book series: Studies in Theoretical and Applied Statistics ((STASSPSS))

  • 1703 Accesses

Abstract

Many models with latent structure are just semi-algebraic sets, and have recently begun to be studied from this perspective; this has shed much light on the dimension, identifiability, and asymptotic statistical properties of these models. Though most of the attention has been on equality constraints, some progress has also been made on evaluating inequalities which might be used to test such models. However, the mathematical complexity of these approaches seems to have led to a gap between our theoretical understanding and the manner in which these models are applied in practice. In this paper we make a plea for some focus on finding simpler (in particular more graphical) and more computationally feasible ways to express such constraints, even at the cost of a loss of statistical power. Recent advances for directed acyclic graph models with latent variables and phylogenetic models are given as illustrations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Allman, E.S., Rhodes, J.A.: Phylogenetic ideals and varieties for the general Markov model. Adv. Appl. Math. 40(2), 127–148 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allman, E.S., Rhodes, J.A., Sturmfels, B., Zwiernik, P.: Tensors of nonnegative rank two (2013). arXiv:1305.0539. arXiv preprint

    Google Scholar 

  3. Allman, E.S., Rhodes, J.A., Taylor, A.: A semialgebraic description of the general Markov model on phylogenetic trees (2012). arXiv:1212.1200. arXiv preprint

    Google Scholar 

  4. Bartolucci, F., Forcina, A.: A likelihood ratio test for MTP2 within binary variables. Ann. Stat. 28(4), 1206–1218 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonet, B.: Instrumentality tests revisited. In: Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, San Francisco, CA, pp. 48–55 (2001)

    Google Scholar 

  6. Cavender, J.A., Felsenstein, J.: Invariants of phylogenies in a simple case with discrete states. J. Classif. 4(1), 57–71 (1987)

    Article  MATH  Google Scholar 

  7. Crow, S.J., Swanson, S.A., Peterson, C.B., Crosby, R.D., Wonderlich, S.A., Mitchell, J.E.: Latent class analysis of eating disorders: relationship to mortality. J. Abnorm. Psychol. 121(1), 225–231 (2012)

    Article  Google Scholar 

  8. Didelez, V., Sheehan, N.: Mendelian randomization as an instrumental variable approach to causal inference. Stat. Methods Med. Res. 16(4), 309–330 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Drton, M., Sturmfels, B., Sullivant, S.: Algebraic factor analysis: tetrads, pentads and beyond. Probab. Theory Relat. Fields 138(3), 463–493 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Drton, M.: Likelihood ratio tests and singularities. Ann. Stat. 37(2), 979–1012 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Evans, R.J.: Graphical methods for inequality constraints in marginalized DAGs. In: IEEE International Workshop on Machine Learning for Signal Processing (2012)

    Book  Google Scholar 

  12. Fienberg, S.E., Hersh, P., Rinaldo, A., Zhou, Y.: Maximum Likelihood Estimation in Latent Class Models for Contingency Table Data, Chap. 2, pp. 27–62. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  13. Glymour, M.M., Tchetgen, E.J.T., Robins, J.M.: Credible Mendelian randomization studies: approaches for evaluating the instrumental variable assumptions. Am. J. Epidemiol. 175(4), 332–339 (2012)

    Article  Google Scholar 

  14. Holland, P.W., Rosenbaum, P.R.: Conditional association and unidimensionality in monotone latent variable models. Ann. Stat. 14(4), 1523–1543 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kang, C., Tian, J.: Inequality constraints in causal models with hidden variables. In: Proceedings of the Conference on Uncertainty in Artificial Intelligence, pp. 233–240, Cambridge, AUAI Press (2006)

    Google Scholar 

  16. Mezuk, B., Kendler, K.S.: Examining variation in depressive symptoms over the life course: a latent class analysis. Psychol. Med. 42, 2037–2046 (2012)

    Article  Google Scholar 

  17. Pearl, J.: On the testability of causal models with latent and instrumental variables. In: UAI-95, pp. 435–443 (1995)

    MathSciNet  Google Scholar 

  18. Ramsahai, R.R., Lauritzen, S.L.: Likelihood analysis of the binary instrumental variable model. Biometrika 98(4), 987–994 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shpitser, I., Evans, R.J., Richardson, T.S., Robins, J.M.: Introduction to nested Markov models. Behaviormetrika, 41(1) 3–39 (2014)

    Article  Google Scholar 

  20. Silva, R., Ghahramani, Z.: The hidden life of latent variables: Bayesian learning with mixed graph models. J. Mach. Learn. Res. 10, 1187–1238 (2009)

    MATH  MathSciNet  Google Scholar 

  21. Tian, J.: Studies in causal reasoning and learning. Ph.D. thesis, UCLA (2002)

    Google Scholar 

  22. Ver Steeg, G., Galstyan, A.: A sequence of relaxations constraining hidden variable models. In: Proceedings of the Twenty-seventh Conference on Uncertainty in Artificial Intelligence, pp. 717–727 (2011)

    Google Scholar 

  23. Verma, T., Pearl, J.: Causal networks: semantics and expressiveness. In: Schachter R., Levitt T.S., Kanal L.N., (eds.) Uncertainty in Artificial Intelligence 4. New York: Elsevier, pp. 69–76 (1990)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Oxford, Oxford, UK

    Robin J. Evans

Authors
  1. Robin J. Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robin J. Evans .

Editor information

Editors and Affiliations

  1. Dept. of Economics and Management, University of Brescia, Brescia, Italy

    Maurizio Carpita

  2. Dept. of Economics and Management, University of Brescia, Brescia, Italy

    Eugenio Brentari

  3. Dept. of Chemometrics and Sensometrics, Oniris Nantes National College, Nantes, France

    El Mostafa Qannari

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Evans, R.J. (2014). Graphical Latent Structure Testing. In: Carpita, M., Brentari, E., Qannari, E. (eds) Advances in Latent Variables. Studies in Theoretical and Applied Statistics(). Springer, Cham. https://doi.org/10.1007/10104_2014_10

Download citation

Publish with us

Policies and ethics

Search

Navigation