Bayesian thermodynamics, measurements and the arrow of time

Cosma Shalizi tries to prove that Bayesian Statistical Mechanics implies a decreasing entropy, and hence a reversed arrow of time. I cannot follow all the formalism, but the gist of it seems to be as follows:

1. Performing a measurement on the system reduces its entropy (paragraph above Eq. (2)).
2.  This contradicts the fact that "in reality, thermodynamic entropy is monotonically
   non-decreasing".
3. Hence, one cannot identify thermodynamic entropy with subjective uncertainty.
4.  Fortunately, since otherwise the theory has some absurd consequences: "watching a pot closely enough [would] keep it from boiling."

He does not address an obvious objection: a system that we keep on measuring is not closed, so point 2. does not apply. To put it differently, Shalizi's observer (or is it the concept of observer he imputes to Bayesian statistics?) is a version of Maxwell's demon (this has already been discussed, see for instance: 1, 2).

We can then accept that measurements reduce the entropy of the observed system. Can we quantify this reduction for current experimental techniques? Is it proportional to the amount of information effectively obtained?