Nuclear polarizability in muonic atoms: Bayesian analysis of the $\eta$-expansion uncertainty

Over the last decade, spectroscopy of muonic atoms has been used to extract nuclear charge radii with unprecedented precision. This extraction requires a careful analysis of the interplay between theoretical and experimental results, and the attainable precision is nowadays limited by the large uncertainties associated with the theoretical evaluation of the nuclear polarizability effects. To facilitate calculations, these polarizability corrections are conventionally expressed as expansions in a dimensionless parameter $\eta$. Based on the uncertainty principle, $\eta$ has been argued to hold a value of approximately 0.33 in light-nuclear systems. In this work, we check this claim by performing a Bayesian analysis of the nuclear-polarizability corrections to the Lamb shift in $\mu{}^2$H and $\mu{}^3$H atoms and in $\mu{}^3$He$^+$ and $\mu{}^4$He$^+$ ions at various orders in the $\eta$-expansion. We derive the Bayesian posterior probability distributions for the values of $\eta$, check for their sensitivity to several reasonable choices of the Bayesian priors, and estimate the uncertainties due to truncation of the $\eta$-expansion. This analysis supports the claim that $\eta\ll1$ for these systems. Overall, the 68\% credible intervals are in good agreement with the simpler truncation uncertainty estimates previously reported in the literature, with the only exception of the $\mu {}^3$He$^+$ ion, for which we find larger uncertainties.