Entropy: An Introduction
homes.cs.washington.edu
homes.cs.washington.edu
An observation/suggestion. The intro is accessible to many people; that drops off a steep cliff when you hit the maths. Now, I'm not complaining about that: it's instructive and necessary to formalise things. Where I struggle is in reading the equations in my head when I don't know what words to use for the symbols. For example, that very first `X ~ p(x)`. I didn't know what to say for the tilde character, so couldn't verbalise the statement. I do know that $\in$ (the rounded 'E') means 'is a member of' so I could read the next statement. The problem gets even more confusing for a non-mathematician as the same symbol is used with different meaning in different branches of maths/science (e.g. $\Pi$).
I get that writing out every equation in English isn't feasible (or, at least, is asking a lot of the writer). But I wonder if there's middle way, e.g. through hyperlinking?
As I say: not a criticism and I don't have a good solution. Just an observation from a non-mathematician. Enjoyed the piece anyway.
Are you sure it's a matter of knowing what to say (in your head) vs knowing the definition of the notation in the first place? I am pretty familiar with this notation, but I rarely verbalize it mentally. I can tell because I read and understand it quickly without problem, but on the rare occasion when I have to read it aloud I realize I'm not sure how I should pronounce it.
Agree it's more "say in my head" than "speak out loud". But I still need to know what to say - internally or externally. Without knowing that ~ denotes "drawn from", all I can say is "X tilde p of x". That has no semantics; no intuition. Whereas knowing that $\in$ means "is a member of", I can read "x \in X" as "x is a member of X".
> but I rarely verbalize it mentally
Neither do I when I know something well. For example, I don't explicitly verbalise "is a member of" now, even internally. There's a shortcut hard-wired in that understands it without needing to pronounce it explicitly. In fact that short cut goes beyond the syntax: it goes straight to the intuition of "x represents any member of the set X". But I had to go through the process of saying it on the way to the shortcut.
Something like X ~ p(x) would be seen all over the place in probability, stats, ML, and related courses such as info theory, detection and estimation, etc. Likely by the time someone is interested in info theory this notation would be permanently etched into their minds. So for this article it is very "audience appropriate".
> not a criticism and I don't have a good solution
Having a mental map of how different subjects fit together (without actually having to studying them in-depth) is a good start.
I've seen so many people crash and burn with machine learning because they were unaware that it depends on linear algebra, calculus, and probability.
With a mental map there is less "surprise" and it's more a matter of simply understanding that they didn't have the right dependencies.
In particular the condition isn't only that it's the shortest code, it's the shortest self-delimiting code. In your example with probabilities {1/2, 1/4, 1/8, 1/8}, someone could come in and say let's code it as {0,1,01,00}, which would appear to encode the latter two outcomes 2 bits rather than 3. The problem, of course, is that {0,1,01,00} is not a self-delimiting code: after you receive the bit 0, you don't know if you're done or if you should wait for another bit to form 01. But the code {0,10,110,111} is self-delimiting, because after you get a 0 or a 10, etc., you know you're done.
I've found that when I teach this material, if I don't mention the self-delimiting condition, then a clever student in the class always thinks of the {0,1,01,00}-type code. (This can be a good way to identify clever students in an intro information theory class!)
The Shannon definition is equivalent to the Boltzmann def only in the case that the micro state consists of infinitely many identical subsystems. If there are only finitely many, for instance, the log of the quantity does not correspond to the same "-p log p".
The Clausius def can be derived from the Boltzmann one, but they are nevertheless also distinct formulations.
According to Wikipedia, if you start with the Gibbs entropy (which is the same as Shannon entropy), and then assume all microstate probabilities are equal (which Boltzmann does), you get the Boltzmann entropy formula. It also says Boltzmann himself used a p ln(p) formulation.
So aren't they the same, perhaps up to a constant factor?
The article does a decent job at graphing and laying out some of the concepts of entropy for information theory, but I'm not sure who the target reader is, since prereqs are perhaps only slightly narrower than what one needs to read Shannon's paper[0] and the article is really illustrating only a fraction of the concept.
It can perhaps work as a primer for what shows up starting on pages 10-11 of the original document, in any case, provided you grasp the mathematical definition of entropy through thermodynamics, and the microstates-based definition through Boltzman, as well as "basic probabilities" (expected value, typical discrete distributions, terms like "i.i.d"), you should be good to go. But then you might already know all this..
And if you do, and you like what you read, then the full original thing by Shannon is a delight to explore to truly grasp what has been so foundational to a lot of things since 1948.
[0] http://people.math.harvard.edu/~ctm/home/text/others/shannon...
https://mechaelephant.com/dev/Shannon-Entropy/
[1] https://homes.cs.washington.edu/~ewein/blog/2020/07/14/entro...
pq^(k-1)log(q)
(k-1)pq^(k-1)log(q)
Great piece, this is really making this concept more intuitive for me.
Not that it wasn't a great article though :).