A precise determination of parton distribution functions (PDFs) is essential not only in furthering our understanding of the composition of stable matter, but also as PDF predictions play an important part in the calculation of Standard Model observables and Beyond the Standard Model searches, and will be even more crucial in light of the high-precision data expected from the high luminosity LHC run. While the Monte Carlo replica method and the Hessian calculation are widely used in PDF determination, the need for a more robust uncertainty treatment has made Bayesian statistics emerge as a desirable alternative. Yet computational cost and the lack of suitable PDF parametrisations have made the realisation of a fully Bayesian PDF fit a challenging endeavour. In this talk, I will present a linear parametrisation that is tailor-made for Bayesian PDF fitting. I will discuss the features of this model that allow us to utilise the Bayesian statistical toolbox, as well as the challenges that still need to be overcome to have the first fully Bayesian PDF fit.