Collaborators:
None.
Key words:
estimation, invariance, parameterization, manifold, metric,
Bayesian, MAP, MMSE,
mean, continuous.
Resume:
It is frequently stated that the maximum a posteriori
(MAP) and minimum mean squared error (MMSE) estimates of a continuous
parameter are not invariant to arbitrary "reparameterizations'' of
the parameter space. This report 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 provides
a solution. We first show that the presence of a metric structure on the
parameter space can be used to define coordinate-invariant MAP and MMSE
estimates, and we argue that this is the natural and necessary way to
proceed. The estimation problem and related geometrical quantities are all
defined in a manifestly coordinate-invariant way. We show that the same
MAP estimate results from 'density maximization' or from using an
invariantly-defined delta function loss. We then discuss the choice of a
metric structure on parameter space. 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. The resulting MAP estimate coincides with the minimum message
length (MML) estimate, but no discretization or approximation is used in
its derivation.
Publications:
"Invariant Bayesian Estimation on Manifolds",
Ian H. Jermyn. To appear in the Annals of Statistics. (PDF)
"On Bayesian Estimation in Manifolds", Ian H.
Jermyn. INRIA Research Report 4607. (PS) |