A solution would be attempts at measuring those parameters directly for a large number of disks (not possible at the moment, except for a few cases and few variables) or use population synthesis calculations, to plug all the planet formation physics we think we understand into a model and see what exoplanet populations it produces.
Those populations are then tested against the real ones, see e.g. Ida & Lin (2008) or Mordasini et al. (2012) (see also the other many papers by the Bern group) and more recent formulations including pebble accretion by Ndugu et al. (2018).
I don't think Bayesian inference is very widely used. The usualy approach is forward Monte-Carlo modelling. Bayesian inference simply wouldn't help that much, as many part of planet formation suffer from significant degeneracies and uncertainties and multiple pathways.
As an example, for post-formation atmospheric evolution, two competing scenarios exist to explain the Fulton (2017) radius gap. Both these scenarios possess different formulae of atmospheric mass-loss. Using a hierarchical model as in e.g. Rogers et al. (2021), one can then infer e.g. the birth-size distribution from the observed size distribution for both formulae. But this works, because it is a relatively simple analysis, in terms of the dimensionality of parameter space.
But in planet formation, there are multiple processes at work, with unknown efficiencies and parameters (dust growth, dust opacities, angular momentum transport, planetesimal formation, oligarchic growth, gas accretion, migration) which all depend in nontrivial ways on the disk parameters. And then it is not guaranteed that the early stages of planet formation happen in a clean-cut disk etc.
So any attempt at Bayesian inference there would be prone to enormous biases and the result of inference would just leave the entirety of parameter space open.