  We propose a statistical approach for estimating the mean line width in
spectra comprising Lorentzian, Gaussian, or Voigt line shapes. Our approach
uses Gaussian processes in two stages to jointly model a spectrum and its
Fourier transform. We generate statistical samples for the mean line width by
drawing realizations for the Fourier transform and its derivative using Markov
chain Monte Carlo methods. In addition to being fully automated, our method
enables well-calibrated uncertainty quantification of the mean line width
estimate through Bayesian inference. We validate our method using a simulation
study and apply it to an experimental Raman spectrum of $\beta$-carotene.
