Can I rant on how bad much mathematical notation is in terms of usability? On the first page, we have:
> A discrete random variable X is completely defined by the set of values it can take, X , which we assume to be a finite set, and its probability distribution {pX(x)}x∈X . The value pX(x) is the probability that the random variable X takes the value x.
Actually, we don't have that, because copy/pasting it here loses essential information. This paragraph uses "X" to refer to three entirely different concepts, distinguished only by the font used.
Not only that, but it relies on subscripting and case, yet uses letters like "p" and "x" which difference in case is mainly in their vertical size and position.
They are not distinguished by font: there is an an upper-case "X" (to mean a random variable) and a lower-case "x" to mean a sample from that radom variable. This is a very reasonable and useful case convention.
Finally there is the Greek letter Chi to mean the set over which that self-same RV varies. Chi seems deliberately chosen because it resembles "X". That might not be a good idea, but there is a defensible argument for it:
The domain Chi is a piece of mathematical machinery that is logically needed, but is not very important to the main disucssion. He wants to note it down, but also let it fade into the background. It's precesly because you want to pay too much attention to it that you are getting upset.
(I'm more worried that he seems to elide the distinction between the domain of an RV and the underlying sample-space. But then, sample-spaces are also mathematical machinery that aren't needed once the rubber hits the road.)
But they're not three entirely different concepts. Regular-capital-X, the random variable, is the object. Italic-X is its range, and lowercase-x is often something like an iterator over that range.
It might be a little clunky in isolation, but it lets you talk about multiple random variables and their properties in a fairly compact way.
You could write X, codomain(X), and possibleValue(X), but this would be pretty verbose.
Once you've famiarized yourself with the notation, which is standard, it's easy to understand and very efficient.
The set script-X, the random variable X, and the value x refer to three different but interrelated concepts. Using entirely different letters would likely make it harder to understand.
I'm not making any claims about whether the notation is good or bad, but just because something is the standard doesn't mean it's therefore good.
And the person you're replying to seems to be well aware it is a standard notation: "Can I rant on how bad much mathematical notation is in terms of usability?"
I am interested in studying Computer Science Theory for the Information Age by Hopcrot and Kanaan[0]. How do these books compare? Is there much overlap between the two? Are they totally different?
Very different. Their text focuses on the connections between stochastic systems, information theory, and machine learning/discrete models in mathematics, whereas the one you linked is purely about statistics and machine learning from a classical perspective.
btw, you've linked to a 2012 version of the book. Since 2012, they've added another author and changed the name of the book. I have no idea how much the content changed.
It looks deep but unmotivated. That doesn't mean there's no motivation, but I'd like it communicated to me more clearly: why should I value the material of this book being put together all in one place?
I'm personally struggling to find a way of answering this that doesn't sound ridiculous on account of the fact that I've been obsessing over this stuff for several decades. These subjects address the heart of what it means to be able to make decisions and use language effectively. This subject matter is essentially a continuation of a broad attempt at a modern conceptualization of meaning, and as such it's value is immediately obvious to me... You should value this material because it is this kind of material that describes what 'value' is, why it is important, and what limitations it has.
But this is all absurd when I try to apply it to a specific book. It's a low rung on a ladder so tall nobody has seen the top.
>I'm personally struggling to find a way of answering this that doesn't sound ridiculous on account of the fact that I've been obsessing over this stuff for several decades.
Then go ahead and sound ridiculous! You've marinated your mind in the material, so I can only benefit from knowing what connections you see.
>These subjects address the heart of what it means to be able to make decisions and use language effectively. This subject matter is essentially a continuation of a broad attempt at a modern conceptualization of meaning, and as such it's value is immediately obvious to me... You should value this material because it is this kind of material that describes what 'value' is, why it is important, and what limitations it has.
Really? I might have used similar descriptions for certain applications of probabilistic modelling and information theory to neuroscience and cognitive science, but why for this particular conjunction of graphical models, randomized forms of NP-complete computational problems, and statistical physics?
>But this is all absurd when I try to apply it to a specific book.
Oh. Well, can you just spout off in general for me, then?
>It's a low rung on a ladder so tall nobody has seen the top.
Well, nobody would call it research if they knew exactly what they were doing.
> A discrete random variable X is completely defined by the set of values it can take, X , which we assume to be a finite set, and its probability distribution {pX(x)}x∈X . The value pX(x) is the probability that the random variable X takes the value x.
Actually, we don't have that, because copy/pasting it here loses essential information. This paragraph uses "X" to refer to three entirely different concepts, distinguished only by the font used.
Not only that, but it relies on subscripting and case, yet uses letters like "p" and "x" which difference in case is mainly in their vertical size and position.
Alright, rant over.