I was trying to understand the equation 8.11 to 8.12 of this paper, where 8.11 suggested:
$H(x, y) = H(x) + H((y|x) = H(x) + H(y - f(x)|x) = H(x) + H(n|x) =H(x,n)$
After searching, I can't find any prof that suggests $H(y|x) = H(y - f(x)|x)$, I'm a bit confused.
Also puzzled why $H(n) = H(\frac{n}{\sigma_n})+ \log \sigma_n$, is this because: $H(\alpha x) = H(x) + \log \alpha$, and consequently this hold true: $H(x) = H(\frac{x}{\alpha}) + \log \alpha$ ?
Have a look at deep and strong definitions of the Entropy.
It turns out that Entropy is usually something like a logarithm. What logarithm depends on the domain of knowledge. Theorists usually like and intend to deal with general cases. Consequently, they avoid concretizing the logarithm. There is some convention in Theory to prefer functional definitions. Function definition in this case is really closed to infinitely differentiable functions that only pose certain properties.
This is brilliantly confirmed by the Wikipedia article example I already gave.
So to add some physical that changes nothing and correct this change with a summand it has to be
$H(\frac{x}{\alpha})-(-log(\alpha))$
following the lay of the logarithm.
To simplify equations for better readability plenty of authors use short form and miss expectation symbols in their equations. They rely on the intelligence, knowledge of the reader to take the expectation values for themselves. A common argument is that there are several applications for the formula and they are going to cover any but the reader may do so.
This applies to equation chains as well. There may be several stages or layers hidden in the given equations chain. One is a restriction to several parts of the system or the chain of what is happening to the system under consideration. There are even convention in the advanced text that formula should not be taken too seriously.
There are initiatives from several groups of authors, reviewers, publishers to write, permit or publish texts that are readable by bigger groups of consumers. But as is seen that are only plans for a future. Writing fast and publishing fast is of importance. The interaction with the author by readers may than lead occasionally to improvements.
Since these are math formulas and math is an exact science the fundamental cognitive blast can not be avoided or overcome. The clause is that the author's intentions are not too serious for his attended audience to go into more details and the details may be found elsewhere for example in advanced courses led by the author.
Many languages are different to English and so there may be even more questions arising from foreign language speakers than the ones the author intended. The theme - Rhema coercions are always hard to solve even if one is restricted to the sure set of methodologies. There is a moment of creativity especially in statistics, thermodynamics, or information or knowledge-related sciences arising from these problems.
There is at least the problem of typesetting. We are restricted to MathJax basic tutorial and quick reference. So the $\vert$ may have very different meanings. I MathJax it is only vertical stroke. The article referred to above uses for example braces with lower indices for the thermodynamic restriction. These are for example constant volume and constant temperature or constant pressure or a constant number of some of the constituents. This might apply to a partial component of a mixture and so on. There are many restrictions possible.