  We recently developed a Bayesian framework for parameter estimation in
general effective field theories. Here we present selected results from using
that framework to estimate parameters with a nucleon-nucleon (NN) potential
derived using chiral effective field theory ($\chi$EFT): the semi-local NN
potential of Epelbaum, Krebs, and Mei{\ss}ner (EKM). There are many NN
scattering data, up to high energies, and with rather small errors, so imposing
a penalty for unnatural low-energy constants (LECs) usually has a small effect
on the fits. In contrast, we have found that including an estimate of higher
orders in $\chi$EFT plays an important role in robust parameter estimation.We
present two case studies where our Bayesian machinery illuminates physics
issues. The first involves the EKM potential at fourth order in the $\chi$EFT
expansion: the two-dimensional posterior probability density function (pdf) for
the fourth-order $s$-wave LECs obtained from the Nijmegen PWA93 phase shifts
indicates these parameters in the NN potential are degenerate. We trace this
feature of the pdf to the presence of an operator in the fourth-order NN
potential that vanishes on-shell. The second case study examines the stability
of LEC extractions as more data at higher energies are included in the fit. We
show that as long as $\chi$EFT truncation errors are properly accounted for in
the parameter estimation, the LEC values extracted using our Bayesian approach
are not sensitive to the maximum energy chosen for the fit.Uncorrelated and
fully correlated models for the truncation errors are compared, pointing the
way to the use of Gaussian processes to more generally model the correlation
structure.
