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Conjugate priors and posterior inference for the matrix Langevin distribution on the Stiefel manifold. (English) Zbl 1459.62238

Summary: Directional data emerges in a wide array of applications, ranging from atmospheric sciences to medical imaging. Modeling such data, however, poses unique challenges by virtue of their being constrained to non-Euclidean spaces like manifolds. Here, we present a unified Bayesian framework for inference on the Stiefel manifold using the Matrix Langevin distribution. Specifically, we propose a novel family of conjugate priors and establish a number of theoretical properties relevant to statistical inference. Conjugacy enables translation of these properties to their corresponding posteriors, which we exploit to develop the posterior inference scheme. For the implementation of the posterior computation, including the posterior sampling, we adopt a novel computational procedure for evaluating the hypergeometric function of matrix arguments that appears as normalization constants in the relevant densities.

MSC:

62R30 Statistics on manifolds
62H10 Multivariate distribution of statistics
62H11 Directional data; spatial statistics
62F15 Bayesian inference
62P10 Applications of statistics to biology and medical sciences; meta analysis

Software:

movMF; BayesDA; rstiefel
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References:

[1] Absil, P.-A., Mahony, R., and Sepulchre, R. (2009). Optimization algorithms on matrix manifolds. Princeton University Press. · Zbl 1147.65043
[2] Bhatia, R. (2009). Positive definite matrices, volume 24. Princeton university press. · Zbl 1133.15017
[3] Brooks, S. P. and Gelman, A. (1998). “General methods for monitoring convergence of iterative simulations.” Journal of Computational and Graphical Statistics, 7(4): 434-455.
[4] Butler, R. W. and Wood, A. T. (2003). “Laplace approximation for Bessel functions of matrix argument.” Journal of Computational and Applied Mathematics, 155(2): 359-382. · Zbl 1027.65033
[5] Casella, G. and Berger, R. L. (2002). Statistical Inference, volume 2. Duxbury Pacific Grove, CA. · Zbl 0699.62001
[6] Chikuse, Y. (1991a). “Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold.” Journal of Multivariate Analysis, 39(2): 270-283. · Zbl 0739.62041
[7] Chikuse, Y. (1991b). “High dimensional limit theorems and matrix decompositions on the Stiefel manifold.” Journal of Multivariate Analysis, 36(2): 145-162. · Zbl 0724.62056
[8] Chikuse, Y. (1998). “Density estimation on the Stiefel manifold.” Journal of Multivariate Analysis, 66(2): 188-206. · Zbl 1130.62322
[9] Chikuse, Y. (2003). “Concentrated matrix Langevin distributions.” Journal of Multivariate Analysis, 85(2): 375-394. · Zbl 1016.62065
[10] Chikuse, Y. (2012). Statistics on Special Manifolds, volume 174. Springer Science & Business Media. · Zbl 1026.62051
[11] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, convexity, and applications. Elsevier. · Zbl 0646.62008
[12] Diaconis, P. and Ylvisaker, D. (1979). “Conjugate priors for exponential families.” The Annals of Statistics, 7(2): 269-281. · Zbl 0405.62011
[13] Doss, C. R. and Wellner, J. A. (2016). “Mode-constrained estimation of a log-concave density.” arXiv preprint arXiv:1611.10335.
[14] Downs, T., Liebman, J., and Mackay, W. (1971). “Statistical methods for vectorcardiogram orientations.” Vectorcardiography, 2: 216-222.
[15] Downs, T. D. (1972). “Orientation statistics.” Biometrika, 665-676. · Zbl 0269.62027
[16] Edelman, A., Arias, T. A., and Smith, S. T. (1998). “The geometry of algorithms with orthogonality constraints.” SIAM Journal on Matrix Analysis and Applications, 20(2): 303-353. · Zbl 0928.65050
[17] Frank, E. (1956). “An accurate, clinically practical system for spatial vectorcardiography.” Circulation, 13(5): 737-749.
[18] Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2014). Bayesian Data Analysis, volume 2. CRC press Boca Raton, FL. · Zbl 1279.62004
[19] Gelman, A., Rubin, D. B., et al. (1992). “Inference from iterative simulation using multiple sequences.” Statistical Science, 7(4): 457-472. · Zbl 1386.65060
[20] Gross, K. I. and Richards, D. S. P. (1987). “Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions.” Transactions of the American Mathematical Society, 301(2): 781-811. · Zbl 0626.33010
[21] Gross, K. I. and Richards, D. S. P. (1989). “Total positivity, spherical series, and hypergeometric functions of matrix argument.” Journal of Approximation Theory, 59(2): 224-246. · Zbl 0692.33010
[22] Gupta, R. D. and Richards, D. S. P. (1985). “Hypergeometric functions of scalar matrix argument are expressible in terms of classical hypergeometric functions.” SIAM Journal on Mathematical Analysis, 16(4): 852-858. · Zbl 0615.33002
[23] Gutiérrez, R., Rodriguez, J., and Sáez, A. (2000). “Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials.” Electronic Transactions on Numerical Analysis, 11: 121-130. · Zbl 0965.33002
[24] Heidelberger, P. and Welch, P. D. (1981). “A spectral method for confidence interval generation and run length control in simulations.” Communications of the ACM, 24(4): 233-245.
[25] Heidelberger, P. and Welch, P. D. (1983). “Simulation run length control in the presence of an initial transient.” Operations Research, 31(6): 1109-1144. · Zbl 0532.65097
[26] Herz, C. S. (1955). “Bessel functions of matrix argument.” The Annals of Mathematics, 474-523. · Zbl 0066.32002
[27] Hill, R. D. and Waters, S. R. (1987). “On the cone of positive semidefinite matrices.” Linear Algebra and its Applications, 90: 81-88. · Zbl 0615.15008
[28] Hobert, J. P., Roy, V., and Robert, C. P. (2011). “Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling.” Statistical Science, 26(3): 332-351. · Zbl 1246.60095
[29] Hoff, P. D. (2009). “Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data.” Journal of Computational and Graphical Statistics, 18(2): 438-456.
[30] Hornik, K. and Grün, B. (2013). “On conjugate families and Jeffreys priors for von Mises-Fisher distributions.” Journal of Statistical Planning and Inference, 143(5): 992-999. · Zbl 1259.62012
[31] Hornik, K. and Grün, B. (2014). “movMF: An R package for fitting mixtures of von Mises-Fisher distributions.” Journal of Statistical Software, 58(10): 1-31.
[32] Ibragimov, I. A. (1956). “On the composition of unimodal distributions.” Theory of Probability & Its Applications, 1(2): 255-260.
[33] Ifantis, E. and Siafarikas, P. (1990). “Inequalities involving Bessel and modified Bessel functions.” Journal of Mathematical Analysis and Applications, 147(1): 214-227. · Zbl 0709.33003
[34] James, A. T. (1964). “Distributions of matrix variates and latent roots derived from normal samples.” The Annals of Mathematical Statistics, 475-501. · Zbl 0121.36605
[35] James, I. M. (1976). The Topology of Stiefel Manifolds, volume 24. Cambridge University Press. · Zbl 0337.55017
[36] Jupp, P. and Mardia, K. (1980). “A general correlation coefficient for directional data and related regression problems.” Biometrika, 163-173. · Zbl 0426.62035
[37] Jupp, P. E. and Mardia, K. V. (1979). “Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions.” The Annals of Statistics, 599-606. · Zbl 0406.62012
[38] Khare, K., Pal, S., Su, Z., et al. (2017). “A Bayesian approach for envelope models.” The Annals of Statistics, 45(1): 196-222. · Zbl 1367.62174
[39] Khatri, C. and Mardia, K. (1977). “The von Mises-Fisher matrix distribution in orientation statistics.” Journal of the Royal Statistical Society. Series B (Methodological), 95-106. · Zbl 0356.62044
[40] Koev, P. and Edelman, A. (2006). “The efficient evaluation of the hypergeometric function of a matrix argument.” Mathematics of Computation, 75(254): 833-846. · Zbl 1117.33007
[41] Kristof, W. (1969). “A theorem on the trace of certain matrix products and some applications.” ETS Research Report Series, 1969(1).
[42] Lin, L., Rao, V., and Dunson, D. (2017). “Bayesian nonparametric inference on the Stiefel manifold.” Statistica Sinica, 27: 535-553. · Zbl 1362.62071
[43] Lui, Y. and Beveridge, J. (2008). “Grassmann registration manifolds for face recognition.” Computer Vision-ECCV 2008, 44-57.
[44] Mardia, K. and Khatri, C. (1977). “Uniform distribution on a Stiefel manifold.” Journal of Multivariate Analysis, 7(3): 468-473. · Zbl 0377.62029
[45] Mardia, K. V. and Jupp, P. E. (2009). Directional Statistics, volume 494. John Wiley & Sons. · Zbl 0935.62065
[46] Mardia, K. V., Taylor, C. C., and Subramaniam, G. K. (2007). “Protein bioinformatics and mixtures of bivariate von Mises distributions for angular data.” Biometrics, 63(2): 505-512. · Zbl 1136.62082
[47] Muirhead, R. J. (1975). “Expressions for some hypergeometric functions of matrix argument with applications.” Journal of Multivariate Analysis, 5(3): 283-293. · Zbl 0312.62042
[48] Muirhead, R. J. (2009). Aspects of multivariate statistical theory, volume 197. John Wiley & Sons. · Zbl 0556.62028
[49] Nagar, D. K., Morán-Vásquez, R. A., and Gupta, A. K. (2015). “Extended matrix variate hypergeometric functions and matrix variate distributions.” International Journal of Mathematics and Mathematical Sciences, 2015. · Zbl 1476.62042
[50] Newton, M. A. and Raftery, A. E. (1994). “Approximate Bayesian Inference with the Weighted Likelihood Bootstrap.” Journal of the Royal Statistical Society. Series B (Methodological), 56(1): 3-48. · Zbl 0788.62026
[51] Pal, S. Sengupta, S., Mitra, R., and Banerjee, A. (2019). “Supplementary material: Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold.” Bayesian Analysis.
[52] Pearson, J. W., Olver, S., and Porter, M. A. (2017). “Numerical methods for the computation of the confluent and Gauss hypergeometric functions.” Numerical Algorithms, 74(3): 821-866. · Zbl 1360.33009
[53] Rao, V., Lin, L., and Dunson, D. B. (2016). “Data augmentation for models based on rejection sampling.” Biometrika, 103(2): 319-335. · Zbl 07072114
[54] Schwartzman, A. (2006). “Random ellipsoids and false discovery rates: Statistics for diffusion tensor imaging data.” Ph.D. thesis, Stanford University.
[55] Sei, T., Shibata, H., Takemura, A., Ohara, K., and Takayama, N. (2013). “Properties and applications of Fisher distribution on the rotation group.” Journal of Multivariate Analysis, 116(Supplement C): 440-455. · Zbl 1283.60011
[56] Turaga, P., Veeraraghavan, A., and Chellappa, R. (2008). “Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision.” In Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, 1-8. IEEE.
[57] van Dyk, D.
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