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Asymmetry
The concept of relative entropy was first discovered by two ponies, Kite Back and Leafblower, and is thus sometimes known as KL Divergence. Most first-year students at CSGU are surprised to learn that an idea so fundamental to their magical studies was developed by two pegasi.
Not that they were racist or anything! It was just that, well, pegasi weren't exactly known for their theoretical mathematics, what with their outdoor predilections and their lack of telekinesis for the sheer amount of writing needed for the advanced stuff. And considering how KL Divergence is mostly used in analyzing communications channels, it's not unreasonable to think that it would've been created by unicorns trying to optimize their sending spells.
But, if one were to ask a pegasus (after explaining what KL Divergence even was because, y'know, most of them are outside clearing clouds rather than studying in the library) if they thought it was odd that one of their own had developed a concept so theoretical, they'd laugh in your face and say that of course it took a pegasus to point out something completely obvious, because the unicorns were too busy with their noses in a book to look up at the sky.
See, KL Divergence is a measure of the distance between two probability distributions. It's denoted as D(p||q), where we are measuring the distance from one probability distribution p to another probability distribution q. Like any distance metric, D(p||q) is equal to zero if and only if p and q are the same "point" in probability space, and if its value is large, then q is very far away from p.
Pegasi, as creatures of the sky, find probability very natural to them. Weather is chaos, but even in chaos there are patterns, and pegasi live or die by playing the odds with these patterns. Navigating a storm is little more than rolling the dice and making sure you remembered to weight them properly beforehoof. Most unicorns accept this as the reason why KL Divergence was first developed by two pegasi.
It is not the only reason, though.
One of the more unique properties of the KL Divergence as a distance metric is as follows:
That is, the distance of p from q according to KL Divergence is not always the same as the distance of q from p. This fact takes students at CSGU years to wrap their heads around and, even after graduating, most of them just write it off as an unintuitive quirk of a very useful measure. Sure, it's not actually possible for distance to be asymmetric, but if the equations work, then there's no harm in assuming that's how it is. It's the same kind of willful suspension of disbelief needed to survive Quantum Physics.
But the pegasi know that it's perfectly possible for distances to change depending on how you measure them. If the unicorns would take their eyes off their books and look up, then they'd see, among the intricate patterns of wind and water in the sky, that there are shapes that look like clouds molded in the shape of creatures, dogs and snakes and octopi and stingrays dancing about high above the mountaintops.
They are not clouds.
And the pegasi choose to stay in the skies, to live in the clouds that float higher than Canterlot itself, because they know that distances can be deceiving.
When the pegasi fly above their fellow wingless ponies, they are close to those things that swim through the air, that have wandered the heavens since the beginning of Time. And those things are far away from them.
And when their colleagues in their schools choose to stay inside, in their homes situated on the ground, they are far away from those that seek out flesh and blood and life. But the clouds-that-are-not-clouds are close to them, so close to their prey in ways that they could never understand and that they won't ever listen to the pegasi about when they try to warn them of the dangers that lurk above.
Because as the limit of D(p||q) approaches zero, p and q must take on the same values.
And every year, there are more clouds in the sky, shaped like dogs and snakes and octopi and stingrays.
And ponies.
Not that they were racist or anything! It was just that, well, pegasi weren't exactly known for their theoretical mathematics, what with their outdoor predilections and their lack of telekinesis for the sheer amount of writing needed for the advanced stuff. And considering how KL Divergence is mostly used in analyzing communications channels, it's not unreasonable to think that it would've been created by unicorns trying to optimize their sending spells.
But, if one were to ask a pegasus (after explaining what KL Divergence even was because, y'know, most of them are outside clearing clouds rather than studying in the library) if they thought it was odd that one of their own had developed a concept so theoretical, they'd laugh in your face and say that of course it took a pegasus to point out something completely obvious, because the unicorns were too busy with their noses in a book to look up at the sky.
See, KL Divergence is a measure of the distance between two probability distributions. It's denoted as D(p||q), where we are measuring the distance from one probability distribution p to another probability distribution q. Like any distance metric, D(p||q) is equal to zero if and only if p and q are the same "point" in probability space, and if its value is large, then q is very far away from p.
Pegasi, as creatures of the sky, find probability very natural to them. Weather is chaos, but even in chaos there are patterns, and pegasi live or die by playing the odds with these patterns. Navigating a storm is little more than rolling the dice and making sure you remembered to weight them properly beforehoof. Most unicorns accept this as the reason why KL Divergence was first developed by two pegasi.
It is not the only reason, though.
One of the more unique properties of the KL Divergence as a distance metric is as follows:
∃p,q: D(p||q) ≠ D(q||p)
That is, the distance of p from q according to KL Divergence is not always the same as the distance of q from p. This fact takes students at CSGU years to wrap their heads around and, even after graduating, most of them just write it off as an unintuitive quirk of a very useful measure. Sure, it's not actually possible for distance to be asymmetric, but if the equations work, then there's no harm in assuming that's how it is. It's the same kind of willful suspension of disbelief needed to survive Quantum Physics.
But the pegasi know that it's perfectly possible for distances to change depending on how you measure them. If the unicorns would take their eyes off their books and look up, then they'd see, among the intricate patterns of wind and water in the sky, that there are shapes that look like clouds molded in the shape of creatures, dogs and snakes and octopi and stingrays dancing about high above the mountaintops.
They are not clouds.
And the pegasi choose to stay in the skies, to live in the clouds that float higher than Canterlot itself, because they know that distances can be deceiving.
When the pegasi fly above their fellow wingless ponies, they are close to those things that swim through the air, that have wandered the heavens since the beginning of Time. And those things are far away from them.
And when their colleagues in their schools choose to stay inside, in their homes situated on the ground, they are far away from those that seek out flesh and blood and life. But the clouds-that-are-not-clouds are close to them, so close to their prey in ways that they could never understand and that they won't ever listen to the pegasi about when they try to warn them of the dangers that lurk above.
Because as the limit of D(p||q) approaches zero, p and q must take on the same values.
And every year, there are more clouds in the sky, shaped like dogs and snakes and octopi and stingrays.
And ponies.
Pics
Author, I feel spoiled. Existential dread and mathematics? Now I've got a whole afternoon of research on KL Divergence to look forward to! (For the curious: yes, it's a real thing.)
My penchant for people putting maths into stories aside, I love this sort of thing. Prose that builds emotion, particularly dread, without characters is something I love and really want to see more of. Who needs stories? This entry proves that not all fiction does.
There's some real cleverness to this piece, too, in the way that it uses our understanding of the natures of pegasi and unicorns. I adore the idea that these complex ideas come so naturally to pegasi entirely because of their different outlook on the world, and I think that was really well executed. There's not a huge amount I could ask this piece to do better—perhaps it could start building that dread earlier, to get the most out of the wordcount—but I do admit to coming out of it and not being fully certain what it was I was meant to take away. The emotion of that final line is clear, but (and I've read through the piece twice now) I can't say that the meaning is. I think that takes away from the strength of that emotion, for me.
You might be losing a few people with all the maths-words, but to me, at least, they add to the immersion of the piece. But I'm used to casually throwing jargon around, so that's hardly a surprise. There's some tonal inconsistencies, too, which isn't a wholly bad thing (I like the idea that someone can write a piece with all this serious, academic text, and balance it out with some prose that's less formal), but I feel that in particular the jump in tone between the first two paragraphs comes a little too early. Would it stand out more if we'd had more time to get used to the original tone? Honestly, I'm not sure, but it's something that could be worth investigating, if you haven't already.
This was a really fun take on the prompt and the pic it chose, and I enjoyed reading it a lot! Well-constructed, clean, clear prose which succeeds (albeit not as well as it could) at its goal is always a delight, and this one had more than a couple of lines that had me smiling at their pretty words. I liked this, author; thank you for writing it.
P.S.
Never has a line of fiction spoken more clearly to my interests ❤
My penchant for people putting maths into stories aside, I love this sort of thing. Prose that builds emotion, particularly dread, without characters is something I love and really want to see more of. Who needs stories? This entry proves that not all fiction does.
There's some real cleverness to this piece, too, in the way that it uses our understanding of the natures of pegasi and unicorns. I adore the idea that these complex ideas come so naturally to pegasi entirely because of their different outlook on the world, and I think that was really well executed. There's not a huge amount I could ask this piece to do better—perhaps it could start building that dread earlier, to get the most out of the wordcount—but I do admit to coming out of it and not being fully certain what it was I was meant to take away. The emotion of that final line is clear, but (and I've read through the piece twice now) I can't say that the meaning is. I think that takes away from the strength of that emotion, for me.
You might be losing a few people with all the maths-words, but to me, at least, they add to the immersion of the piece. But I'm used to casually throwing jargon around, so that's hardly a surprise. There's some tonal inconsistencies, too, which isn't a wholly bad thing (I like the idea that someone can write a piece with all this serious, academic text, and balance it out with some prose that's less formal), but I feel that in particular the jump in tone between the first two paragraphs comes a little too early. Would it stand out more if we'd had more time to get used to the original tone? Honestly, I'm not sure, but it's something that could be worth investigating, if you haven't already.
This was a really fun take on the prompt and the pic it chose, and I enjoyed reading it a lot! Well-constructed, clean, clear prose which succeeds (albeit not as well as it could) at its goal is always a delight, and this one had more than a couple of lines that had me smiling at their pretty words. I liked this, author; thank you for writing it.
P.S.
It's the same kind of willful suspension of disbelief needed to survive Quantum Physics.
Never has a line of fiction spoken more clearly to my interests ❤
I wasn't sold on this being much of a story until the ending. I like it but I have a couple of points to make.
First off, even I wouldn't go quite this technical, and I wrote Broken Symmetry for buck's sake. I don't think every paragraph should include the words "KL Divergence" or the story should get as technical as it does with probability and communication theory. I'd try to play it to a non-academic audience and use analogy more than direct definitions.
Second, the ending is good, but it's too short. We're left with a faceless enemy we know almost nothing about, and I think I need a little more here to understand what the danger really is and why I should care. Also, why are the earth ponies more in danger? Is it because they're less imaginative? Is it because they can't see the danger as clearly and the danger is psychological rather than physical? I'm really not sure, and this is the whole point of the story so it needs to be clearer for me.
First off, even I wouldn't go quite this technical, and I wrote Broken Symmetry for buck's sake. I don't think every paragraph should include the words "KL Divergence" or the story should get as technical as it does with probability and communication theory. I'd try to play it to a non-academic audience and use analogy more than direct definitions.
Second, the ending is good, but it's too short. We're left with a faceless enemy we know almost nothing about, and I think I need a little more here to understand what the danger really is and why I should care. Also, why are the earth ponies more in danger? Is it because they're less imaginative? Is it because they can't see the danger as clearly and the danger is psychological rather than physical? I'm really not sure, and this is the whole point of the story so it needs to be clearer for me.
∃p,q: D(p||q) ≠ D(q||p)
"That's a bold strategy, Cotton Candy. Let's see if it pays off for them."
... I learned the hard way, with Hard Reset 2, how polarizing making mathematical equations a fundamental part of your story is going to be. You're probably going to thrill 10 percent of your audience, lose 50 percent of them, and make 10 percent of them upset. I did — and I was writing to an audience that had already survived 12 chapters of nested time loops. So, yeah, I agree upthread: I think the smart play here is to dumb this down to layman's terms. I've got a mathematics degree, and even I had to take a hard break to switch mental modes and parse the story more slowly and carefully — you probably lost most of your non-technical readers.
Also agreed that the ending redeems this. More detail might be nice, but on the other hoof, that lingering sense of unexplained impossibility does its job pretty effectively And I'll throw in on the side of those for whom the tonal shifts didn't really work: this tries so hard to be dry and textual when it's dry and textual that the casual asides threw me off.
I do like what this is aiming at, and it does hit some of those targets squarely. Thanks for the experiment! And as a consolation prize, I'll just note that the aforementioned HR2 is much less dead than it looks, since you seem like the sort of person who might appreciate knowing that. ^..^
Also, huge props for what >>QuillScratch notes about basing KL divergence off of a real thing!
Tier: Flawed but Fun
For what it's worth, I don't know any math more advanced than the most basic algebra, and I got it just fine. I ignored all the complicated stuff, and allowed this line:
to do all the heavy lifting for me.
Really liked the mixing of the scientific and the divine here. Basically echoing Quill. The renaming/repurposing of the real-life KL Divergence. Reminds me of how in TRON: Uprising, the protagonist is named Beck, and one of the central antagonists Cyrus – referencing the real-life Cyrus-Beck algorithm.
That is, the distance of p from q according to KL Divergence is not always the same as the distance of q from p.
to do all the heavy lifting for me.
Really liked the mixing of the scientific and the divine here. Basically echoing Quill. The renaming/repurposing of the real-life KL Divergence. Reminds me of how in TRON: Uprising, the protagonist is named Beck, and one of the central antagonists Cyrus – referencing the real-life Cyrus-Beck algorithm.
I'm a dummy and ended up tuning out ninety percent of the mathematical portions. While I found the ending, its implications, and the larger message appropriately chilling... I think most of the storytelling sailed over my head.
This is a soft abstain. I liked it, but I couldn't follow its contents well enough to judge it fairly as a whole product. I hope you forgive me, author, for weaseling out.
This is a soft abstain. I liked it, but I couldn't follow its contents well enough to judge it fairly as a whole product. I hope you forgive me, author, for weaseling out.