A new analysis technique for Balmer line spectroscopy that enables recombination rate (particle sink) and ionization rate (particle source) inference in a closed divertor configuration is reported. Bayesian inference is employed to systematically utilize all available information from multiple Balmer lines and constrain parameter ranges by using prior knowledge about plasmas. While a closed-divertor facilitates detachment, neutral plugging typically leads to large spatial variations in plasma parameters. A forward model is developed to take into account non-uniformity in the plasma parameters and applied to test data generated by divertor plasma simulations. It is shown that the forward model robustly provides particle source and sink inference over a wide parameter range. In addition, the precision improves as more Balmer lines are resolved simultaneously. The new analysis technique is also applied to an L-mode ASDEX Upgrade plasma in the high-recycling regime. The inferred quantities and their profiles are consistent with the expectations of a high-recycling divertor plasma. The further insight into the detachment physics will be provided by using this new analysis technique.