Some caveats should be considered with the above.
1.  Like all statistical mechanical results according to the MaxEnt school, this increase in thermodynamic entropy is only a prediction. It assumes in particular that the initial macroscopic description contains all of the information relevant to predicting the later macroscopic state. This may not be the case, for example if the initial description fails to reflect some aspect of the preparation of the system which later becomes relevant. In that case the "failure" of a MaxEnt prediction tells us that there is something more which is relevant that we may have overlooked in the physics of the system.
It is also sometimes suggested that quantum measurement, especially in the decoherence interpretation, may give an apparently unexpected reduction in entropy per this argument, as it appears to involve macroscopic information becoming available which was previously inaccessible. (However, the entropy accounting of quantum measurement is tricky, because to get full decoherence one may be assuming an infinite environment, with an infinite entropy).
2.  The argument so far has glossed over the question of fluctuations. It has also implicitly assumed that the uncertainty predicted at time t for the variables at time t will be much smaller than the measurement error. But if the measurements do meaningfully update our knowledge of the system, our uncertainty as to its state is reduced, giving a new S which is less than S. (Note that if we allow ourselves the abilities of Laplaces demon, the consequences of this new information can also be mapped backwards, so our uncertainty about the dynamical state at time t is now also reduced from S to S').
We know that S > S; but we can now no longer be certain that it is greater than S = S. This then leaves open the possibility for fluctuations in S. The thermodynamic entropy may go "down" as well as up. A more sophisticated analysis is given by the entropy Fluctuation Theorem, which can be established as a consequence of the time-dependent MaxEnt picture.
3.  As just indicated, the MaxEnt inference runs equally well in reverse. So given a particular final state, we can ask, what can we "retrodict" to improve our knowledge about earlier states?  However the Second Law argument above also runs in reverse: given macroscopic information at time t, we should expect it too to become less useful.  The two procedures are time-symmetric. But now the information will become less and less useful at earlier and earlier times. (Compare with Loschmidt's paradox.) The MaxEnt inference would predict that the most probable origin of a currently low-entropy state would be as a spontaneous fluctuation from an earlier high entropy state. But this conflicts with what we know to have happened, namely that entropy has been increasing steadily, even back in the past.
The MaxEnt proponents' response to this would be that such a systematic failing in the prediction of a MaxEnt inference is a "good" thing. It means that there is thus clear evidence that some important physical information has been missed in the specification the problem. If it is correct that the dynamics "are" time-symmetric, it appears that we need to put in by hand a prior probability that initial configurations with a low thermodynamic entropy are more likely than initial configurations with a high thermodynamic entropy. This cannot be explained by the immediate dynamics. Quite possibly, it arises as a reflection of the evident time-asymmetric evolution of the universe on a cosmological scale (see arrow of time).