Name: Jari P. Kaipio


CompanyName: University of Kuopio

Country: FINLAND

Abstract: Optimality of current patterns with
respect to posterior covariance measures

The theory of optimal current patterns that
is based on Isaacson's distinguishability
criterion has been proved to be very
successful. In the framework of statistical
interpretation of inverse problems this
criterion is related to maximizing the
difference between the associated likelihood
densities. Although regularization is usually
employed and we strictly speaking can not
refer to the likelihood, this approach works
very well in most such situations that are
usually considered. However, with respect to
the theory of statistical inverse problems,
criteria that focus on likelihood densities
are not relevant. Rather, the final aim is
to consider posterior densities and measures.
In this paper we formulate the optimal current
pattern problem in the statistical inverse
problems setting by considering optimality
with respect to measures on posterior
covariance. These measures are directly
related to estimation error, that is, the
accuracy of the actual conductivity estimates.
The determination of the optimal current
patterns is not based on the actual (or
guessed) conductivity distribution of the
target but rather on the prior density of the
target. We discuss several examples and show
that the optimal current patterns that
correspond to the distinguishability criterion
are usually good also with respect to the
posterior criteria especially when the prior
covariances are diagonal and the reference
conductivity distribution is known. However,
with more complex prior densities it can be
difficult or impossible to construct the two
densities that the distinguishability criterion
is based on. However, for theses densities
of classes the determination of the optimal
current patterns with respect to the posterior
measures is straightforward.

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