Let's say you're a company that's providing an internet connection to a business. The company trusts you, so there's only compression of bits over the wire, not encryption, and you're aware of the compression scheme the company is using to send their bits to you. You're charging the company a premium for using the line you manage but you also lease the line, so it's in your interest to compress what they give you as best as possible so as to make a profit.
Say the companies compression scheme is imperfect. They have a Huffman coding of their (imperfect) model of tokens they send, call it q(x) (that is, they think token x shows up with probability q(x)). You've determined the true distribution, p(x) (token x shows up with actual probability p(x)).
The business has tokens that show up with probability p(x) but they encode them with lg(q(x)) bits, giving an average token bit size of:
-\sum _ x p(x) lg(q(x))
If you then use an optimal Huffman encoding, you will send tokens with average bit length of:
-\sum _ x p(x) lg(p(x))
How many bits, on average, do you save? Just the difference:
-\sum _ x p(x) lg(p(x)) - \sum _ x p(x) lg(q(x)) = -\sum _ x p(x) lg(p(x)/q(x))
Which is the Kullback-Leibler divergence.
To me, this is a much more intuitive explanation. I made a blog post about it [0], if anyone cares.
To rephrase what you wrote in plain English: you are Amazon, a client uses an S3 bucket to store .zip files in them, which they pay by the byte, you re-compress and store the data as .7z files, and the KL divergence is related to zip_file_size - 7z_file_size, your "win".
Wow this is really great. I just realised last weak that MLE can be motivated with the KL divergence between true distribution and approximation. My mind was blown in how obvious that connection was.
Unfortunately all these intuitions rely on a distinction between a "true" distribution P and a "false" distribution Q. So they don't work for a subjective probability interpretation where it doesn't make sense to speak of a true or false distribution.
I wouldn't say it's gentle but it certainly is a great book. Great exercise problems. Some of the proofs are so elegantly done, especially the way calculus of variations is avoided.
David Mackay's book hand holds a little more than Cover and Thomas, although it's remit is more than just information theory.
For those wondering where is this practically relevant - this is the basic metric used to compare quantization of various LLM models - what is the KL divergence of a 4-bit quantization versus an 8 bit one versus the original 16 bit one.
> D(P||Q) = measure of how much our model Q differs from the true distribution P. In other words, we care about how much P and Q differ from each other in the world where P is true, which explains why KL-div is not symmetric.
I don't think this particular interpretation actually makes sense or would explain why KL divergence is not symmetric.
First of all, the "difference" between P and Q would be the same independently of whether P, Q, or some other distribution is the "true" distribution.
For example, assume we have a coin and P(Heads)=0.4 and Q(Heads)=0.6. Now the difference between the two distributions is clearly the same irrespective of whether P, Q or neither is "true". So this interpretation doesn't explain why the KL divergence is asymmetric.
Second, there are plausible cases where it arguably doesn't even make sense to speak of a "true" distribution in the first place.
For example, consider the probability that there was once life on Mars. Assume P(Life)=0.4 and Q(Life)=0.6. What would it even mean for P to be "true"? P and Q could simply represent the subjective beliefs of two different people, without any requirement of assuming that one of these probabilities could be "correct".
Clearly the KL divergence can still be calculated and presumably sensibly interpreted even in the subjective case. But the interpretations in this article don't help us here since they require objective probabilities where one distribution is the "true" one.
Here's my explanation:
Let's say you're a company that's providing an internet connection to a business. The company trusts you, so there's only compression of bits over the wire, not encryption, and you're aware of the compression scheme the company is using to send their bits to you. You're charging the company a premium for using the line you manage but you also lease the line, so it's in your interest to compress what they give you as best as possible so as to make a profit.
Say the companies compression scheme is imperfect. They have a Huffman coding of their (imperfect) model of tokens they send, call it q(x) (that is, they think token x shows up with probability q(x)). You've determined the true distribution, p(x) (token x shows up with actual probability p(x)).
The business has tokens that show up with probability p(x) but they encode them with lg(q(x)) bits, giving an average token bit size of:
If you then use an optimal Huffman encoding, you will send tokens with average bit length of: How many bits, on average, do you save? Just the difference: Which is the Kullback-Leibler divergence.To me, this is a much more intuitive explanation. I made a blog post about it [0], if anyone cares.
[0] https://mechaelephant.com/dev/Kullback-Leibler-Divergence.ht...
To rephrase what you wrote in plain English: you are Amazon, a client uses an S3 bucket to store .zip files in them, which they pay by the byte, you re-compress and store the data as .7z files, and the KL divergence is related to zip_file_size - 7z_file_size, your "win".
Wow this is really great. I just realised last weak that MLE can be motivated with the KL divergence between true distribution and approximation. My mind was blown in how obvious that connection was.
Holy over the top almighty. Is this comment even real ? "Mind blown" and all. Tomorrow, the sun rose, "blown is my mind".
Apologies for the snark but I can't fathom how someone who is aware of the definition of KL not see the likelihood in it.
This is great. I had only ever seen the expected surprise explanation. The others help to fill in the gaps.
Unfortunately all these intuitions rely on a distinction between a "true" distribution P and a "false" distribution Q. So they don't work for a subjective probability interpretation where it doesn't make sense to speak of a true or false distribution.
Is there a gentler intro to this topic?
Try the textbook Elements of Information Theory by Cover and Thomas (2006)
I wouldn't say it's gentle but it certainly is a great book. Great exercise problems. Some of the proofs are so elegantly done, especially the way calculus of variations is avoided.
David Mackay's book hand holds a little more than Cover and Thomas, although it's remit is more than just information theory.
For those wondering where is this practically relevant - this is the basic metric used to compare quantization of various LLM models - what is the KL divergence of a 4-bit quantization versus an 8 bit one versus the original 16 bit one.
> D(P||Q) = measure of how much our model Q differs from the true distribution P. In other words, we care about how much P and Q differ from each other in the world where P is true, which explains why KL-div is not symmetric.
I don't think this particular interpretation actually makes sense or would explain why KL divergence is not symmetric.
First of all, the "difference" between P and Q would be the same independently of whether P, Q, or some other distribution is the "true" distribution.
For example, assume we have a coin and P(Heads)=0.4 and Q(Heads)=0.6. Now the difference between the two distributions is clearly the same irrespective of whether P, Q or neither is "true". So this interpretation doesn't explain why the KL divergence is asymmetric.
Second, there are plausible cases where it arguably doesn't even make sense to speak of a "true" distribution in the first place.
For example, consider the probability that there was once life on Mars. Assume P(Life)=0.4 and Q(Life)=0.6. What would it even mean for P to be "true"? P and Q could simply represent the subjective beliefs of two different people, without any requirement of assuming that one of these probabilities could be "correct".
Clearly the KL divergence can still be calculated and presumably sensibly interpreted even in the subjective case. But the interpretations in this article don't help us here since they require objective probabilities where one distribution is the "true" one.