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# New Riemannian Priors on the Univariate Normal Model

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

* Author to whom correspondence should be addressed.

Received: 17 April 2014 / Revised: 23 June 2014 / Accepted: 9 July 2014 / Published: 17 July 2014

(This article belongs to the Special Issue Information Geometry)

# Abstract

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*{*

*p*

*θ*;

*θ*

*∈*Θ

*}*is any parametrization of the univariate normal model, the paper considers prior distributions

*G*($\stackrel{-}{\theta}$

*, γ*) with hyperparameters $\stackrel{-}{\theta}$

*∈*Θ and

*γ*

*>*0, whose density with respect to Riemannian volume is proportional to exp(

*−*

*d*2(

*θ, $\stackrel{-}{\theta}$)*where d

*/*2*γ*^{2}),^{2}(θ, $\stackrel{-}{\theta}$) is the square of Rao’s Riemannian distance. The distributions G($\stackrel{-}{\theta}$, γ) are termed Gaussian distributions on the univariate normal model. The motivation for considering a distribution G($\stackrel{-}{\theta}$, γ) is that this distribution gives a geometric representation of a class or cluster of univariate normal populations. Indeed, G($\stackrel{-}{\theta}$, γ) has a unique mode $\stackrel{-}{\theta}$(precisely, $\stackrel{-}{\theta}$ is the unique Riemannian center of mass of G($\stackrel{-}{\theta}$, γ), as shown in the paper), and its dispersion away from $\stackrel{-}{\theta}$ is given by γ. Therefore, one thinks of members of the class represented by G($\stackrel{-}{\theta}$, γ) as being centered around $\stackrel{-}{\theta}$ and lying within a typical distance determined by γ. The paper defines rigorously the Gaussian distributions G($\stackrel{-}{\theta}$, 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($\stackrel{-}{\theta}$, 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.

*Keywords:*Fisher information; Riemannian metric; prior distribution; univariate normal distribution; image classification

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*