  We recently developed an Effective Field Theory (EFT) for rotational bands in
odd-mass nuclei. Here we use EFT expressions to perform a Bayesian analysis of
data on the rotational energy levels of $^{99}$Tc, ${}^{155,157}$Gd,
${}^{159}$Dy, ${}^{167, 169}$Er, ${}^{167, 169}$Tm, ${}^{183}$W, ${}^{235}$U
and ${}^{239}$Pu. The error model in our Bayesian analysis includes both
experimental and EFT truncation uncertainties. It also accounts for the fact
that low-energy constants (LECs) at even and odd orders are expected to have
different sizes. We use Markov Chain Monte Carlo (MCMC) sampling to explore the
joint posterior of the EFT and error-model parameters and show both the LECs
and the expansion parameter, $Q$, can be reliably determined. We extract the
LECs up to fourth order in the EFT and find that, provided we correctly account
for EFT truncation errors in our likelihood, results for lower-order LECs are
stable as we go to higher orders. LEC results are also stable with respect to
the addition of higher-energy data. We extract the expansion parameter for all
the nuclei listed above and find a clear correlation between the extracted and
the expected $Q$ based on the single-particle and vibrational energy scales.
However, the $Q$ that actually determines the convergence of the EFT expansion
is markedly smaller than would be naively expected based on those scales.
