Does the ergodic hypothesis provide a uniquely determined definition of entropy?

Question

One can distinguish between two schools of thought regarding thermodynamic entropy:

(a) Thermodynamic entropy is a measure of the "amount of hidden information" in a system. Therefore, the entropy depends on which information we consider to be "hidden". (If we have god-like powers of observation and we can see the position and velocity of every particle, then the entropy is zero).

(b) Assuming that the ergodic hypothesis holds, then the thermodynamic entropy is the logarithm of the phase space volume which an ergodic system explores. Therefore, entropy is completely determined by the physics.

Are these two points of view fundamentally incompatible? Which one gives the correct experimental predictions?

Note: this is slightly different from other questions about whether the entropy is subjective, such as

Entropy : subjective lack of knowledge that leads to objective conclusions

There is an "objective" way to define entropy in school (a), namely the hidden information is the one we are unable to access. The question is whether the thermodynamic entropy thus depends on our capabilities (which is not the same as depending on our beliefs).

Answer 1

One argument (still hoping for a more satisfying one) is that whatever information we have access to about the system should not be time-dependent in thermal equilibrium (otherwise it wouldn't be equilibrium!) It follows the most information about the system that we can reasonably have access to is that it lies somewhere on an ergodic surface. And since we can find out which ergodic surface the system is occupying by measuring macroscopic thermodynamic variables, this is also the least information about the system we can have access to if we are not being deliberately ignorant. Therefore there is only really one good definition of thermodynamic entropy, even if we define it from point of view (a).

However, Jaynes does have some interesting thought experiments in which entropy depends on whether an internal degree of freedom of atoms is accessible or not, and this has physical consequences:

http://www.damtp.cam.ac.uk/user/tong/statphys/jaynes.pdf

Unfortunately, these involve entropies of mixing, which are hard to treat in classical mechanics without running into paradoxes, at least if you define entropy as phase space volume [school (b)]; Jaynes does argue that these paradoxes disappear in point of view (a), which is an argument in favor of (a). Still, it would be interesting to see if there are other examples.