scholarly journals Global Geometry of Bayesian Statistics

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 240
Author(s):  
Atsuhide Mori
Keyword(s):  
Bayesian Statistics ◽  
Cholesky Decomposition ◽  
Information Geometry ◽  
Covariance Matrices ◽  
Multivariate Normal ◽  
Hamiltonian Flow ◽  
Normal Distributions ◽  
Global Geometry ◽  
Hilbert Modular ◽  
Symplectic Leaves

In the previous work of the author, a non-trivial symmetry of the relative entropy in the information geometry of normal distributions was discovered. The same symmetry also appears in the symplectic/contact geometry of Hilbert modular cusps. Further, it was observed that a contact Hamiltonian flow presents a certain Bayesian inference on normal distributions. In this paper, we describe Bayesian statistics and the information geometry in the language of current geometry in order to spread our interest in statistics through general geometers and topologists. Then, we foliate the space of multivariate normal distributions by symplectic leaves to generalize the above result of the author. This foliation arises from the Cholesky decomposition of the covariance matrices.

A Test for Equality of Means of Multivariate Normal Distributions when Covariance Matrices are Unequal

1971 ◽  
Vol 20 (4) ◽  
pp. 153-156 ◽  
Author(s):  
R. P. Bhargava
Keyword(s):  
Covariance Matrices ◽  
Multivariate Normal ◽  
Normal Distributions ◽  
Normal Populations ◽  
Equality Of Means ◽  
Multivariate Normal Distributions ◽  
Test Of Significance

Summary In this paper a test of significance is given for testing the equality of means of q+ 1 multivariate normal populations ( q > 1) when their covariance matrices are unequal and unknown.

scholarly journals Symplectic/Contact Geometry Related to Bayesian Statistics

Proceedings ◽  
2019 ◽  
Vol 46 (1) ◽  
pp. 13
Author(s):  
Atsuhide Mori
Keyword(s):  
Probability Density Functions ◽  
Convolution Product ◽  
Geometric Description ◽  
Poisson Structures ◽  
Multivariate Normal ◽  
Hamiltonian Flow ◽  
Data Smoothing ◽  
Normal Distributions ◽  
Normal Means ◽  
Single Function

In the previous work, the author gave the following symplectic/contact geometric description of the Bayesian inference of normal means: The space H of normal distributions is an upper halfplane which admits two operations, namely, the convolution product and the normalized pointwise product of two probability density functions. There is a diffeomorphism F of H that interchanges these operations as well as sends any e-geodesic to an e-geodesic. The product of two copies of H carries positive and negative symplectic structures and a bi-contact hypersurface N. The graph of F is Lagrangian with respect to the negative symplectic structure. It is contained in the bi-contact hypersurface N. Further, it is preserved under a bi-contact Hamiltonian flow with respect to a single function. Then the restriction of the flow to the graph of F presents the inference of means. The author showed that this also works for the Student t-inference of smoothly moving means and enables us to consider the smoothness of data smoothing. In this presentation, the space of multivariate normal distributions is foliated by means of the Cholesky decomposition of the covariance matrix. This provides a pair of regular Poisson structures, and generalizes the above symplectic/contact description to the multivariate case. The most of the ideas presented here have been described at length in a later article of the author.

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