Context. Quiescent solar prominence fine structures are typically modelled as density enhancements, called threads, which occupy a fraction of a longer magnetic flux tube. This is justified from the spatial distribution of the imaged plasma emission or absorption of prominences at small spatial scales. The profile of the mass density along the magnetic field is unknown, however, and several arbitrary alternatives are employed in prominence wave studies. The identification and measurement of period ratios from multiple harmonics in standing transverse thread oscillations offer a remote diagnostics method to probe the density variation of these structures. Aims: We present a comparison of theoretical models for the field-aligned density along prominence fine structures. They aim to imitate density distributions in which the plasma is more or less concentrated around the centre of the magnetic flux tube. We consider Lorentzian, Gaussian, and parabolic profiles. We compare theoretical predictions based on these profiles for the period ratio between the fundamental transverse kink mode and the first overtone to obtain estimates for the density ratios between the central part of the tube and its foot-points and to assess which one would better explain observed period ratio data. Methods: Bayesian parameter inference and model comparison techniques were developed and applied. To infer the parameters, we computed the posterior distribution for the density gradient parameter that depends on the observable period ratio. The model comparison involved computing the marginal likelihood as a function of the period ratio to obtain the plausibility of each density model as a function of the observable. We also computed the Bayes factors to quantify the relative evidence for each model, given a period ratio observation. Results: A Lorentzian density profile, with plasma density concentrated around the centre of the tube, seems to offer the most plausible inversion result. A Gaussian profile would require unrealistically high values of the density gradient parameter, and a parabolic density distribution does not enable us to obtain well-constrained posterior probability distributions. However, our model comparison results indicate that the evidence points to the Gaussian and parabolic profiles for period ratios in between 2 and 3, while the Lorentzian profile is preferred for higher period ratio values. The method we present can be used to obtain information on the plasma structure along threads, provided period ratio measurements become widely available.