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IPI

We discuss Bayesian inverse problems in Hilbert spaces. The focus is on a
fast concentration of the posterior probability around the unknown
true solution as expressed in the concept of posterior contraction
rates. This concentration is dominated by a parameter which controls
the variance of the prior distribution. Previous results determine
posterior contraction rates based on known solution smoothness. Here
we show that an oracle-type parameter choice is possible. This is done
by relating the posterior contraction rate to the root mean squared estimation
error. In addition we show that the tail probability, which usually
is bounded by using the Chebyshev inequality, has exponential decay, at
least for

*a priori*parameter choices. These results implement the exponential concentration of Gaussian measures in Hilbert spaces.## Year of publication

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