  Kernels are efficient in representing nonlocal dependence and they are widely
used to design operators between function spaces. Thus, learning kernels in
operators from data is an inverse problem of general interest. Due to the
nonlocal dependence, the inverse problem can be severely ill-posed with a
data-dependent singular inversion operator. The Bayesian approach overcomes the
ill-posedness through a non-degenerate prior. However, a fixed non-degenerate
prior leads to a divergent posterior mean when the observation noise becomes
small, if the data induces a perturbation in the eigenspace of zero eigenvalues
of the inversion operator. We introduce a data-adaptive prior to achieve a
stable posterior whose mean always has a small noise limit. The data-adaptive
prior's covariance is the inversion operator with a hyper-parameter selected
adaptive to data by the L-curve method. Furthermore, we provide a detailed
analysis on the computational practice of the data-adaptive prior, and
demonstrate it on Toeplitz matrices and integral operators. Numerical tests
show that a fixed prior can lead to a divergent posterior mean in the presence
of any of the four types of errors: discretization error, model error, partial
observation and wrong noise assumption. In contrast, the data-adaptive prior
always attains posterior means with small noise limits.
