The complex nature of inertial confinement fusion (ICF) experiments results in a very large number of experimental parameters which, when combined with the myriad physical models that govern target evolution, make the reliable extraction of physics from experimental campaigns very difficult. We develop an inference method that allows all important experimental parameters, and previous knowledge, to be taken into account when investigating underlying microphysics models. The result is framed as a modified χ2 analysis which is easy to implement in existing analyses, and quite portable. We present a first application to a recent convergent ablator experiment performed at the National Ignition Facility (NIF), and investigate the effect of variations in all physical dimensions of the target (very difficult to do using other methods). We show that for well characterized targets in which dimensions vary at the 0.5% level there is little effect, but 3% variations change the results of inferences dramatically. Our Bayesian method allows particular inference results to be associated with prior errors in microphysics models; in our example, tuning the carbon opacity to match experimental data (i.e. ignoring prior knowledge) is equivalent to an assumed prior error of 400% in the tabop opacity tables. This large error is unreasonable, underlining the importance of including prior knowledge in the analysis of these experiments.