He also has a paper on the arXiv that is flat-out wrong, so ignore "The Backwards Arrow of Time of the Coherently Bayesian Statistical Mechanic", though showing how it goes wrong takes a fair bit of explaining of fairly subtle points.

I've tried reading it before - for me to understand just the paper itself would also take a fair bit of explaining of fairly subtle points! I understand Shalizi's sketch of his argument in words:

Observe your system at time 0, and invoke your favorite way of going from an observation to a distribution over the system's states --- say the maximum entropy principle. This distribution will have some Shannon entropy, which by hypothesis is also the system's thermodynamic entropy. Assume the system's dynamics are invertible, so that the state at time t determines the states at times t+1 and t-1. This will be the case if the system obeys the usual laws of classical mechanics, for example. Now let your system evolve forward in time for one time-step. It's a basic fact about invertible dynamics that they leave Shannon entropy invariant, so it's still got whatever entropy it had when you started. Now make a new observation. If you update your probability distribution using Bayes's rule, a basic result in information theory shows that the Shannon entropy of the posterior distribution is, on average, no more than that of the prior distribution. There's no way an observation can make you more uncertain about the state on average, though particular observations may be very ambiguous. (Noise-free measurements would let us drop the "on average" qualifer.) Repeating this, we see that entropy decreases over time (on average).

My problem is that I know a plausible-looking argument expressed in words can still quite easily be utterly wrong in some subtle way, so I don't know how much credence to give Shalizi's argument.

If you write such a post, I'll almost certainly upvote it.