Brazilian Journal of Probability and Statistics

A Bayesian analysis of the Bingham distribution

Stephen G. Walker

Full-text: Open access

Abstract

This paper provides a means, using latent variables, to undertake a Bayesian analysis for the Bingham distribution. To date, this has been problematic due to a nonclosed form for the normalizing constant. Previous approaches have relied on approximating the constant; something which is unnecessary in the method adopted here.

Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 1 (2014), 61-72.

Dates
First available in Project Euclid: 5 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1391611337

Digital Object Identifier
doi:10.1214/12-BJPS193

Mathematical Reviews number (MathSciNet)
MR3165428

Zentralblatt MATH identifier
06291460

Keywords
Latent variables Markov chain Monte Carlo normalizing constant posterior distribution

Citation

Walker, Stephen G. A Bayesian analysis of the Bingham distribution. Braz. J. Probab. Stat. 28 (2014), no. 1, 61--72. doi:10.1214/12-BJPS193. https://projecteuclid.org/euclid.bjps/1391611337


Export citation

References

  • Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. Journal of the Royal Statistical Society, Ser. B 55, 25–37.
  • Bingham, C. (1974). An antipodally symmetric distribution on the sphere. The Annals of Statistics 2, 1201–1205.
  • Damien, P., Wakefield, J. C. and Walker, S. G. (1999). Gibbs sampling for Bayesian nonconjugate and hierarchical models using auxiliary variables. Journal of the Royal Statistical Society, Ser. B 61, 331–344.
  • Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. The Annals of Statistics 33, 1643–1665.
  • Godsill, S. J. (2001). On the relationship between Markov chain Monte Carlo methods for model uncertainty. Journal of Computational and Graphical Statistics 10, 230–248.
  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732.
  • Kent, J. T. (1987). Asymptotic expansions for the Bingham distribution. Applied Statistics 36, 139–144.
  • Kume, A. and Wood, A. T. A. (2005). Saddlepoint approximations for the Bingham and Fisher–Bingham normalizing constant. Biometrika 92, 465–476.
  • Kume, A. and Wood, A. T. A. (2007). On the derivatives of the normalizing constant of the Bingham distribution. Statistics and Probability Letters 77, 832–837.
  • MacEachern, S. N. and Berliner, L. M. (1994). Subsampling the Gibbs sampler. The American Statistician 48, 188–190.
  • Mardia, K. V. (1975). Statistics of directional data. Journal of the Royal Statistical Society, Ser. B 37, 349–393.
  • Mardia, K. V. and Zemroch, P. J. (1977). Maximum likelihood estimators for the Bingham distribution. Journal of Statistical Computation and Simulation 6, 29–34.
  • Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Chichester: Wiley.
  • Meyer, R., Fournier, D. A. and Berg, A. (2003). Stochastic volatility: Bayesian computation using automatic differentiation and the extended Kalman filter. Econometrics Journal 6, 407–419.
  • Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Ser. B 55, 3–23.
  • Walker, S. G. (2011). Posterior sampling when the normalizing constant is unknown. Communications in Statistics 40, 784–792.