Jaynes' Description of Maximum Entropy Distribution So I am reading E. T. Jaynes probability theory book, and I am at chapter 11 where he introduces the maximum entropy principle. I understand that Jaynes separates the notion of probability from that of frequency, which I like. But his arguments leading the the maximum entropy principle seem to mix up these two concepts:
Following the example in the book at section 11.5, let's say that we have a 3 sided "die", which we have thrown for an unknown number of times and we only know the sample mean of the results, called $\overline{m}$. We want to assign a probability to each of the 3 sides of the die. Jaynes reasons that, given only this information, the most reasonable way to assign these probabilities is to choose the most uncertain probability distribution that satisfies all the requirements (i.e. observations, in this case the sample mean). Now, this is quite arbitrary, but I accept it as an intuitively compelling method.
Then, Jaynes reasons that, because we know the sample mean turned out to be $\overline{m}$, this is a constraint for our assigned probability distribution's mean to also be $\overline{m}$. This is where he loses me. If I understand correctly, the sample mean is only a feature of the frequency distribution, and cannot be used to rule out any probability distribution, because, except for some edge cases, any sample mean is possible (i.e. probability of larger than zero) for any probability distribution. So, Probability distributions that do not have mean $\overline{m}$ are also compatible with the observation that the sample mean equals $\overline{m}$ and should be considered in the optimization process (and in this case, if we accept my argument, we have to assign a uniform probability distribution, regardless of the value of $\overline{m}$, as opposed to Jaynes' result).
Is there anything I am missing?
Edit:
Please note that, when I say I accept this method, I mean that I accept that using the most uncertain probability distribution that satisfies constraints is a good measure for assigning probabilities. The part of Jaynes' reasoning that I do not accept is that sample mean is a constraint for the probability mean (as I reasoned above). The reason I do not accept it is that this seems to contradict the whole philosophy that the previous chapters were built upon, which is frequency$\neq$probability in the general case.

Jaynes' reasoning here is essentially Bayesian, where the probability is a measure of uncertainty of our knowledge about something. This is opposed to frequentist reasoning, which is based on the assumption that one could potentially know the whole population, its parameters, and the underlying probability distribution (such knowledge is often impossible - e.g., the population can be infinite, or we might be dealing with a series events that may continue far into the future - like sunrise.)
Thus, Jaynes bases his reasoning on the knowledge available to him (such as sample average $\bar{m}$), but does not make assumptions about what the "true" population average $\mu$ is or whether such a population average even exists.
