Gaussian distributions on the space of symmetric positive definite matrices from Souriau’s Gibbs state for Siegel domains by coadjoint orbit and moment map. (English) Zbl 1486.60024
Nielsen, Frank (ed.) et al., Geometric science of information. 5th international conference, GSI 2021, Paris, France, July 21–23, 2021. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 12829, 245-255 (2021).
Summary: We will introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau’s covariant Gibbs density by considering this space as the pure imaginary axis of the homogeneous Siegel upper half space where Sp (2n,R)/U(n) acts transitively. Gauss density of SPD matrices is computed through Souriau’s moment map and coadjoint orbits. We will illustrate the model first for Poincaré unit disk, then Siegel unit disk and finally upper half space. For this example, we deduce Gauss density for SPD matrices.
For the entire collection see [Zbl 1482.94007].
For the entire collection see [Zbl 1482.94007].
MSC:
60E05 | Probability distributions: general theory |
62B11 | Information geometry (statistical aspects) |
15A63 | Quadratic and bilinear forms, inner products |
82B30 | Statistical thermodynamics |
Keywords:
symmetric positive definite matrices; Lie groups thermodynamics; symplectic geometry; maximum entropy; exponential density familyReferences:
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