Now, Shalizi's point is that if we are strict Bayesians about the state of the system, then the entropy of the distribution we associate with it will not increase, so we would say that the entropy of the system is decreasing. But this is wrong! The entropy of the system is increasing. Not the entropy of the system+observer combo, the entropy of the system itself. If your approach to statistical mechanics tells you it is not, then you are the one flying in the face of orthodox thermodynamics, not Shalizi.
This is the part I take issue with. Everything else is fine. The entropy of the distribution that we associate with the system will decrease, at the expense of pumping our ignorance as waste heat into our mind's surrounding environment. The entropy of my beliefs about the system is not the same thing as the entropy of the system and it's not covered under the umbrella of orthodox physics to act like my state of ignorance regarding the state of the system is the same as the state of the system. When I learn things by pumping entropy into my surroundings with (at the very least) my brain's waste heat, that is not at all like observing a backward arrow of time, because everything else around me is running down, reaching thermal equilibrium, even if I am pinching up some local ignorance-removal regarding the state of some different, fixed other system.
On what basis are you making the claim that my brain pumping heat into its environment is equivalent to my brain pumping entropy into my environment? Can you justify this claim using a purely Bayesian approach to entropy? I can't see how that would work.
Link to the Question
I haven't gotten an answer on this yet and I set up a bounty; I figured I'd link it here too in case any stats/physics people care to take a crack at it.