Geometric mean of probability measures and geodesics of Fisher information metric

Mitsuhiro Itoh and Hiroyasu Satoh

Mathematische Nachrichten, 2023, vol. 296, issue 5, 1901-1927

Abstract: The space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by P(M)$\mathcal {P}(M)$, carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of P(M)$\mathcal {P}(M)$, equipped with G. We show that a geodesic segment joining arbitrary probability measures μ1 and μ2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of P(M)$\mathcal {P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function ℓ defined by ℓ(μ1,μ2):=2arccos∫Mp1p2dλ$\ell \!\big (\mu _1, \mu _2\big ):=2\arccos \int \nolimits _M \sqrt {p_1p_2}\,d\lambda$, μi=piλ$\mu _i=p_i \lambda$, i=1,2$i=1,2$, gives the Riemannian distance function on P(M)$\mathcal {P}(M)$. It is shown that geodesics are all minimal.

Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202000167

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:296:y:2023:i:5:p:1901-1927

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().