Estimating parameters that quantify the physical environments of solar and stellar atmospheres requires a detailed understanding of complex instrumentation and/or atomic physics, both of which can sometimes be modeled as forward convolutions. The weighting functions for these convolutions are themselves the product of precursor statistical analyses and may involve another set of parameters of scientific interest. In practice the errors arising from the precursor analyses are ignored in secondary analyses. We present a Bayesian framework for coherently accounting for these uncertainties in the secondary analysis. This framework allows us to estimate both the physical parameters of interest and the convolution functions. In principle, multiple data sets sharing common convolution functions can be combined for more precise inference. Unfortunately, however, this may allow biases stemming from misspecification in some of the analyses to spread to others. We consider how comparing the individual analyses can diagnose such bias and how the results of the combined secondary analyses can be fed back to improve estimation of their parameters of a precursor analysis.