  Bayesian linear inverse problems aim to recover an unknown signal from noisy
observations, incorporating prior knowledge. This paper analyses a data
dependent method to choose the scale parameter of a Gaussian prior. The method
we study arises from early stopping methods, which have been successfully
applied to a range of problems for statistical inverse problems in the
frequentist setting. These results are extended to the Bayesian setting. We
study the use of a discrepancy based stopping rule in the setting of random
noise. Our proposed stopping rule results in optimal rates under certain
conditions on the prior covariance operator. We furthermore derive for which
class of signals this method is adaptive. It is also shown that the associated
posterior contracts at the optimal rate and provides a conservative measure of
uncertainty. We implement the proposed stopping rule using the continuous-time
ensemble Kalman--Bucy filter (EnKBF). The fictitious time parameter replaces
the scale parameter, and the ensemble size is appropriately adjusted in order
to not lose statistical optimality of the computed estimator. The EnKBF, then,
gives a continuous process from the prior distribution to the posterior which
is terminated using the proposed stopping rule.
