New Riemannian Priors on the Univariate Normal Model

Entropy

Lionel Bombrun¹  Yannick Berthoumieu¹  Salem Said¹ 

[1] Groupe Signal et Image, CNRS Laboratoire IMS, Institut Polytechnique de Bordeaux, Université de Bordeaux, UMR 5218, Talence, 33405, France;

关键词: Fisher information;    Riemannian metric;    prior distribution;    univariate normal distribution;    image classification;   

DOI  :  10.3390/e16074015

来源: DOAJ

【 摘 要 】

The current paper introduces new prior distributions on the univariate normal model, with the aim of applying them to the classification of univariate normal populations. These new prior distributions are entirely based on the Riemannian geometry of the univariate normal model, so that they can be thought of as “Riemannian priors”. Precisely,if {pθ ; θ ∈ Θ} is any parametrization of the univariate normal model, the paper considersprior distributions G(θ - , γ) with hyperparameters θ -∈ Θ and γ > 0, whose density with respect to Riemannian volume is proportional to exp(−d2(θ, θ - )/2γ2), where d2(θ, θ - ) is the square of Rao’s Riemannian distance. The distributions G(θ - , γ) are termed Gaussiandistributions on the univariate normal model. The motivation for considering a distribution G(θ - , γ) is that this distribution gives a geometric representation of a class or cluster of univariate normal populations. Indeed, G(θ - , γ) has a unique mode θ - (precisely, θ -is the unique Riemannian center of mass of G(θ - , γ), as shown in the paper), and its dispersion away from θ -is given by γ.  Therefore, one thinks of members of the class represented by G(θ - , γ) as being centered around θ -and  lying within a typical  distance determined by γ. The paper defines rigorously the Gaussian distributions G(θ - , γ) and describes an algorithm for computing maximum likelihood estimates of their hyperparameters. Based on this algorithm and on the Laplace approximation, it describes how the distributions G(θ - , γ) can be used as prior distributions for Bayesian classification of large univariate normal populations. In a concrete application to texture image classification, it is shown that  this  leads  to  an  improvement  in  performance  over  the  use  of  conjugate  priors.

【 授权许可】

Unknown