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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 |

### Keywords:

Bayesian inference; conjugate prior; hypergeometric function of matrix argument; matrix Langevin distribution; Stiefel manifold; vectorcardiography
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\textit{S. Pal} et al., Bayesian Anal. 15, No. 3, 871--908 (2020; Zbl 1459.62238)

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