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.

scholarly journals A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Dong Hyun Cho
Keyword(s):  
Simple Formula ◽  
Fourier Transforms ◽  
Convolution Product ◽  
Multivariate Normal ◽  
Normal Distributions ◽  
Conditional Expectations ◽  
Change Of Scale ◽  
Convolution Products ◽  
Borel Class ◽  
Generalized Cylinder

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of L2[0,T]. We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

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.

NON-NORMAL STATISTICAL TOLERANCE ANALYSIS USING ANALYTICAL CONVOLUTION METHOD

2008 ◽  
Vol 07 (01) ◽  
pp. 127-130 ◽  
Author(s):  
S. G. LIU ◽  
P. WANG ◽  
Z. G. LI
Keyword(s):  
Normal Distribution ◽  
Closed Loop ◽  
Tolerance Analysis ◽  
Accurate Method ◽  
Probability Density Functions ◽  
Triangular Distribution ◽  
Normal Distributions ◽  
Statistical Tolerance ◽  
Statistical Tolerance Analysis ◽  
Convolution Method

In statistical tolerance analysis, it is usually assumed that the statistical tolerance is normally distributed. But in practice, there are many non-normal distributions, such as uniform distribution, triangular distribution, etc. The simple way to analyze non-normal distributions is to approximately represent it with normal distribution, but the accuracy is low. Monte-Carlo simulation can analyze non-normal distributions with higher accuracy, but is time consuming. Convolution method is an accurate method to analyze statistical tolerance, but there are few reported works about it because of the difficulty. In this paper, analytical convolution is used to analyze non-normal distribution, and the probability density functions of closed loop component are obtained. Comparing with other methods, convolution method is accurate and faster.

Sign in / Sign up

Export Citation Format

Share Document