Open Access
April 2005 Invariant Bayesian estimation on manifolds
Ian H. Jermyn
Ann. Statist. 33(2): 583-605 (April 2005). DOI: 10.1214/009053604000001273


A frequent and well-founded criticism of the maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter γ taking values in a differentiable manifold Γ is that they are not invariant to arbitrary “reparameterizations” of Γ. This paper clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a sine qua non for a mathematically well-defined problem, and diffeomorphism invariance, which is a substantial issue, and then provides a solution. We first show that the presence of a metric structure on Γ can be used to define coordinate-invariant MAP and MMSE estimates, and we argue that this is the natural way to proceed. We then discuss the choice of a metric structure on Γ. By imposing an invariance criterion natural within a Bayesian framework, we show that this choice is essentially unique. It does not necessarily correspond to a choice of coordinates. In cases of complete prior ignorance, when Jeffreys’ prior is used, the invariant MAP estimate reduces to the maximum likelihood estimate. The invariant MAP estimate coincides with the minimum message length (MML) estimate, but no discretization or approximation is used in its derivation.


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Ian H. Jermyn. "Invariant Bayesian estimation on manifolds." Ann. Statist. 33 (2) 583 - 605, April 2005.


Published: April 2005
First available in Project Euclid: 26 May 2005

zbMATH: 1069.62004
MathSciNet: MR2163153
Digital Object Identifier: 10.1214/009053604000001273

Primary: 62A01 , 62C10 , 62F10 , 62F15

Keywords: Bayesian , continuous , estimation , Invariance , Manifold , MAP , ‎mean‎ , metric , MMSE , parameterization

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 2 • April 2005
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