I was wondering about good critiques of Jaynes' approach to statistical mechanics. @Yvan did a good job in pointing out a couple of them, but I would like to have them fleshed out a bit, especially by someone who is not biased towards Jaynes.
As I think Jaynes' thoughts on this matter shifted a bit through the years, let me summarise what I think his position is:
Logically, one must start with two things:
There is a microscopic theory of the phenomenon under consideration --- for the moment that can be the existence of a (quantum or classical) hamiltonian formulation, which then ensures the existence of a preferred Louiville form. Thus it makes sense to discuss probabilities of trajectories (ensemble), independently of where those probabilities come from.
There exists a viable macroscopic, coarse-grained description, which is only the case if an experiment says so --- the key is what Jaynes sometimes calls "reproducibility". If a phenomenon is not readily reproduced then clearly one has not gained sufficient control over enough variables --- e.g. it was an experimental fact that controlling temperature and volume of a gas was sufficient to determine its pressure.
Then it is logically true that one could develop a quantitative theory/relationship of the macroscopic degrees of freedom or observables, and the claim is that one should set up an ensemble over the microscopic trajectories subject to the constraints of the macroscopic observations and a unique one is chosen by maximising entropy.
With the ensemble in hand, one could then proceed to make predictions about other observables, failure to then observe them means either your microscopic theory is wrong, or the set of variables chosen is not correct, and the circle of Science is complete.
Importantly this makes no reference to ergodicity, and in fact this works out of equilibrium --- equilibrated systems just tends to help with experimental reproducibility. Personally I see it as morally dimensional analysis, writ large.
Yvan pointed out that there is a problem with classical hamiltonian systems because they have an (uncountable) underlying configuration space, and there are technical problems with defining entropy on them. My opinion is that this does not matter, because all physical hamiltonians are really quantum, and those come with a canonical choice. (Field theories need regularisation, both UV and IR as always, to render the number of degrees of freedom finite to be "physical".) However, I'm probably just being naive and I would certainly welcome education on this.
My preferred reference (never published, I think): Where do we stand on maximum entropy?