I appreciate your patience with me and for the help in getting me to confront my confusions about the topic. Your answer still is unsatisfying to me, and this could totally be my own ignorance at work. However, I cannot understand how your answer is sustainable given the comments at both the Stack Exchange post and at the John Baez link.

I think you've misunderstood me when you articulated the 3 positions listed above, but you've definitely hit upon my confusion so I need to think about it more carefully and do a better job saying what I want to say. I will think on it and write again when I get a chance this weekend.

Again, I do appreciate the patience in helping me understand it.

"The entropy of the probability distribution describing the system" only has meaning if there is an observer to actually hold that probability distribution. Since probability is in the mind, there is no fixed external thing that just "is" the probability distribution of the system.

There are two distinct things; one is "the system" and the other is "the probability distribution over states of the system." If you make an idealization and do math just on "the system" then the distributions in those idealizations are entropy increasing (if you exclude any observer or external stuff to the system). That does not correspond to reality (because the system's not truly closed), but is often a useful approximation for describing "the arrow of time."

If you want to talk about "the probability distribution over states of the system" then you must also be including some observer with a mind of some sort, or else the notion of there being a probability distribution (as opposed to just whatever the deterministic eventuality of whatever does in fact occur) doesn't make semantic sense.

So, to speak about the "probability distribution of the system" there has to be a Maxwell's demon sitting there having that distribution in its mind (e.g. some observer), and whatever entity it is that is dissipating waste heat while doing physical processes to update its beliefs must be increasing entropy.

This is the semantic problem that you dismissed. When I talk about the refrigerator, it's clear that I mean to draw an imaginary boundary around the refrigerator only and pretend for a second that that is all there is anywhere. Then the entropy is decreasing. If I talk about the process by which I acquired that knowledge, then I have to expand my imaginary boundary to include the source of the photons that bounced off the refrigerator, for instance, and the waste heat my brain produced to acquire this knowledge. That process, the acquiring of the knowledge, was entropy increasing even if what it revealed to me was a less entropic distribution over states of the refrigerator.

The refrigerator is the two gas system with a pump attached. Learning anything about either system is an entropy increasing proposition (if the boundary is drawn around me plus the system). As it happens, if you want to draw the boundary to exclude me, then the two-gas-system-without-pump also happens to be entropy increasing, while drawing a boundary around the refrigerator is entropy decreasing.

This seriously is just Maxwell's demon.

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.

There appears to be a semantic problem with this (I am not a physicist, so please bear with me).

If "the arrow of time" is re-defined to just mean "superficial appearance of decreases in entropy to some observer", then I agree with Shalizi and I also believe the result of his paper is not a 'paradox' and doesn't cast any doubt on validity of Bayesian methods. In local situations, a system might be sufficiently "closed" such that to the observer it looks like the system is spontaneously becoming more complex... that is, the degree of ignorance in the observer's mind might decrease quickly.

But, consistent with the physical laws, somewhere within the observer-system metasystem, that entropy is being accounted for. In order to zoom out and re-apply Shalizi's idea to the meta-system, you have to start talking about some new meta-observer whose states of ignorance are only relevant to the first observer-system metasystem.

So to me, it seems like if your approach accurately describes Shalizi's argument, then all he is doing is redefining "arrow of time" such that he gets the result he wants... but no one has to care about that version of "arrow of time" nor believe that it corresponds to the same "arrow of time" that is discussed in almost all discourse on thermodynamics. And even less should anyone think this is genuine reason to be skeptical of fully Bayesian updating.

Can you articulate how your response interfaces with this answer on Cross-Validated?

I don't understand the disagreement with splitting the reputation. For example, a really trivially easy way to do it would be like this: On every post, have a thumbs-up/thumbs-down vote button that is specific just to that post, and then have a separate thumbs-up/thumbs-down button that appears next to the name of the user who made the post.

If you just dislike that particular post because it is off-topic, but you think the poster had the intention that it was on-topic (you just dispute that they were correct in their intention), then just downvote the question and not the user. Then the user voting is a signal of an individual's favor in the community and the post voting is a signal of the community's preferences for topical content.

I'm not advocating that we go through the trouble of doing it that way, but it would be an easy way to decouple the second order effect by which a user can feel personally discouraged if a post he or she thought was relevant and interesting is not seen that way by others. Their reputation as a contributing member may remain unchanged; but that particular post is signaled as uninteresting/noisy.

I would like a FAQ that functions much the way the guidelines function at the Stack Exchange websites. Without any guidelines, downvotes are chaotic and lose meaning. If a typical user doesn't like a post, but the reason for dislike is not covered by the FAQ, they can still write a comment, or make a post in one of the Stack Exchange meta sites (to argue constructively for getting their preference category into the FAQ/guidelines). These signal the information and successfully decouple it from what the community says it wants in the FAQ.

Sorry, I should have been clearer; I think it is disingenuous to downvote for that reason if there is no FAQ or guideline expressing the current instantiation of the community's preferences for what is on- or off-topic.

I am all for using a numerical mechanism like voting to aggregate information, but it does come with a reputation cost for people who make the effort to create a post or pass on a link. So it does more than just signal what is liked or disliked; it also has a personal element that may discourage people from trying. If the expectations are clearly spelled out, then voting/downvoting is fine and can be interpreted against that informative backdrop.

Also, this sort of voting allows us to aggregate a coarse "yes" or "no" kind of preference about a post, but I think it would be pretty difficult to impute nuanced preferences, such as classifying topics and sub-topics as on-topic or off-topic, just by aggregating these votes. There's no clear delineation of the "why" behind the vote, and that metadata is more important for understanding the squiggly, discontinuous boundary between "on-topic" and "off-topic". Without the "why" metadata, we're getting a workable, but very coarse, low-resolution, smoothed boundary between the two. I advocate that a FAQ or guidelines for downvoting is a low-cost method to raise the resolution of that boundary.

Is there any sort of FAQ or guideline about which topics "are about rationality" and which "are not about rationality"? I think it's a little disingenuous to downvote because it doesn't satisfy your definition of being "about rationality," as opposed to merely writing a comment that says, "I am not sure that this post is sufficiently on topic."

I am also curious how much of the regular LW reading community feels the same way. I would like to restrict top-level posts to being directly about rationality and the topics discussed in the sequences, but I am also glad there is a sort of "overflow" area where the topics of posts are free to have higher variance. I also conjecture that by opening up the Discussion part of LW to more haphazard posts, readers can get a higher sense of affiliation with other readers and this has a positive effect on the effort expended by the marginal reader (at the expense of lowering the site's overall signal-to-noise ratio). My view is that that trade-off is clearly worth it, but you seem to prefer a different view.