So, why should we believe that it works to put probability distributions on functions of both data and parameters, in a non-generative way? (see my previous example)
Well, one reason is the success of Approximate Bayesian Computation (ABC), in that scheme typically you have a physical simulator, like for example a PDE or an ODE solver, or an agent-based model or something, and if you put in the parameters, the simulator will return you a prediction for the data..... and then, you have data. And the data never matches the predictions exactly, and there's no arguments you can make about IID samples, since typically every measurement in space is tied together by the whole simulation. So what do you do?
The typical method is to calculate several statistics of the simulation and data, and see if they fit together close enough. For example you might calculate the total kinetic energy of some ensemble, calculate the mean and stddev of reflectivity of infra-red over your domain, calculate the density of predators in a certain region... whatever is appropriate for your model, you get some kind of summary statistic (or several of them), and you compare it to the data, and you accept the parameters if the difference between the two is in the high probability region of some distribution (could be just if you're within epsilon, in other words, a uniform distribution).
This method works, in lots of examples, and it is a generalized version of what I did in my previous example. Its success gives confidence to the idea that writing "non generative" models whose whole structure simply puts high probability on parameters when some function of the data an parameters is in the high probability region of a particular distribution you've chosen.
This is a pretty powerful generalization from the typical IID likelihood principle!