Statistics and Computing

, Volume 26, Issue 1–2, pp 349–360 | Cite as

Exact Bayesian inference for the Bingham distribution

  • Christopher J. Fallaize
  • Theodore KypraiosEmail author


This paper is concerned with making Bayesian inference from data that are assumed to be drawn from a Bingham distribution. A barrier to the Bayesian approach is the parameter-dependent normalising constant of the Bingham distribution, which, even when it can be evaluated or accurately approximated, would have to be calculated at each iteration of an MCMC scheme, thereby greatly increasing the computational burden. We propose a method which enables exact (in Monte Carlo sense) Bayesian inference for the unknown parameters of the Bingham distribution by completely avoiding the need to evaluate this constant. We apply the method to simulated and real data, and illustrate that it is simpler to implement, faster, and performs better than an alternative algorithm that has recently been proposed in the literature.


Directional statistics Bayesian inference Markov Chain Monte Carlo Doubly intractable distributions 



The authors are most grateful to Richard Arnold and Peter Jupp for providing the earthquake data and John Kent for providing a Fortran program to compute moments of the Bingham distribution. Finally, we would like to thank Ian Dryden for commenting on an earlier draft of this manuscript and Andy Wood for helpful discussions.


  1. Andrieu, C., Thoms, J.: A tutorial on adaptive mcmc. Stat. Comput. 18(4), 343–373 (2008)MathSciNetCrossRefGoogle Scholar
  2. Arnold, R., Jupp, P.E.: Statistics of orthogonal axial frames. Biometrika 100(3), 571–586 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  3. Aune, E., Simpson, D.P., Eidsvik, J.: Parameter estimation in high dimensional gaussian distributions. Stat. Comput. 24, 1–17 (2012)MathSciNetGoogle Scholar
  4. Bingham, C.: An antipodally symmetric distribution on the sphere. Ann. Stat. 2, 1201–1225 (1974)Google Scholar
  5. Boomsma, W., Mardia, K.V., Taylor, C.C., Ferkinghoff-Borg, J., Krogh, A., Hamelryck, T.: A generative, probabilistic model of local protein structure. Proc. Natl. Acad. Sci. USA 105(26), 8932–8937 (2008)CrossRefGoogle Scholar
  6. Caimo, A., Friel, N.: Bayesian inference for exponential random graph models. Soc. Net. 33(1), 41–55 (2011)CrossRefGoogle Scholar
  7. Ehler, M., Galanis, J.: Frame theory in directional statistics. Stat. Probab. Lett. 81(8), 1046–1051 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. Everitt, R.G.: Bayesian parameter estimation for latent markov random fields and social networks. J. Comput. Gr. Stat. 21(4), 940–960 (2012)Google Scholar
  9. Friel, N.: Evidence and bayes factor estimation for gibbs random fields. J. Comput. Gr. Stat. 22(3), 518–532 (2013)MathSciNetCrossRefGoogle Scholar
  10. Ganeiber, A.M.: Estimation and simulation in directional and statistical shape models. PhD thesis, University of Leeds (2012)Google Scholar
  11. Girolami, M., Lyne A.M., Strathmann, H., Simpson, D., Atchade, Y.: Playing russian roulette with intractable likelihoods. ArXiv preprint; arXiv:1306.4032, (2013)
  12. Hamelryck, Thomas: Kanti V Mardia. Jesper Ferkinghoff-Borg. Bayesian methods in structural bioinformatics. Springer-Verlag, Berlin (2012)CrossRefGoogle Scholar
  13. Kent, J.T.: Asymptotic expansions for the Bingham distribution. J. Roy. Stat. Soc. Ser. C 36(2), 139–144 (1987)Google Scholar
  14. Kent, J.T., Ganeiber, A.M., Mardia, K.V.: A new method to simulate the Bingham and related distributions in directional data analysis with applications. ArXiv Preprint, (2013).
  15. Kume, A., Wood, A.T.A.: Saddlepoint approximations for the Bingham and Fisher-Bingham normalising constants. Biometrika 92(2), 465–476 (2005)Google Scholar
  16. Kume, A., Wood, A.T.A.: On the derivatives of the normalising constant of the Bingham distribution. Stat. Probab. Lett. 77(8), 832–837 (2007)Google Scholar
  17. Kume, A., Walker, S.G.: Sampling from compositional and directional distributions. Stat. Comput. 16(3), 261–265 (2006)Google Scholar
  18. Levine, J.D., Funes, P., Dowse, H.B., Hall, J.C.: Resetting the circadian clock by social experience in drosophila melanogaster. Sci. Signal. 298(5600), 2010–2012 (2002)Google Scholar
  19. Mardia, K.V., Zemroch, P.J.: Table of maximum likelihood estimates for the bingham distribution. Statist. Comput. Simul. 6, 29–34 (1977)zbMATHCrossRefGoogle Scholar
  20. Mardia, K.V., Jupp, P.E.: Directional statistics. Wiley Series in probability and statistics. John Wiley & Sons Ltd., Chichester (2000). ISBN 0-471-95333-4Google Scholar
  21. Møller, J., Pettitt, A.N., Reeves, R., Berthelsen, K.K.: An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika 93(2), 451–458 (2006)Google Scholar
  22. Murray, I., Ghahramani, Z., MacKay, D.J.C.: MCMC for doubly-intractable distributions. In Proceedings of the 22nd annual conference on uncertainty in artificial intelligence (UAI-06), pages 359–366. AUAI Press (2006)Google Scholar
  23. Ripley, B.D.: Stochastic simulation. Wiley Series in probability and mathematical statistics: applied probability and statistics. John Wiley & Sons Inc., New York, (1987). ISBN 0-471-81884-4. doi: 10.1002/9780470316726
  24. Rueda, C., Fernández, M.A., Peddada, S.D.: Estimation of parameters subject to order restrictions on a circle with application to estimation of phase angles of cell cycle genes. J. Am. Stat. Assoc. 104(485), 338–347 (2009) Google Scholar
  25. Sei, T., Kume, A.: Calculating the normalising constant of the bingham distribution on the sphere using the holonomic gradient method. Stat. Comput., 1–12 (2013)Google Scholar
  26. Storvik, G.: On the flexibility of Metropolis–Hastings acceptance probabilities in auxiliary variable proposal generation. Scandinavian J. Stat. 38(2), 342–358 (2011)Google Scholar
  27. Tyler, D.E.: Statistical analysis for the angular central Gaussian distribution on the sphere. Biometrika 74(3), 579–589 (1987)Google Scholar
  28. Walker, S.G.: Posterior sampling when the normalizing constant is unknown. Comm. Stat. Simul. Comput. 40(5), 784–792 (2011)Google Scholar
  29. Walker, S.G.: Bayesian estimation of the Bingham distribution. Braz. J. Probab. Stats. 28(1), 61–72 (2014)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity Park, University of NottinghamNottinghamUK

Personalised recommendations