p log q

K-L divergence is the pretty much really dumb, obvious thing you would probably think up if someone asked you to compare two probability distributions.

You would start by going "oh I suppose I will look at how far apart they are at all the different values" and add it up. Then you would say, "oh, but 0.1 and 0.01 are really an order of magnitude apart, just like 0.01 and 0.001. Perhaps it will work better if I use the log of the probability". Then you would pause for thought and say, "Hmmmm hang on some of the values are really extreme but almost never happen, shouldn't it be weighted by how frequently they occur?".

But of course paragraphs of mathematical symbols are the way many people prefer to express this.

I don't think if anyone was asked simply to "compare two probability distributions", they'd come up with something asymmetric like the KL divergence. You'd need to at least add that one of the distributions is the "real" distribution.

Log shows up a lot but I don't think I ever really understood when/how do people decided to use it in their formulas. One KL divergence youtube video said, `let's normalize the p(P)/p(Q) using (log(p(P)/q(P)))^N` and then shows how to derived the KL formula. I'd also appreciate if you know how the N (sample size ig) is being used here.

For a lot of math stuff I endup being confused about how people use operations. Like using 1/p(X=x) is like magic to me because I don't understand what the context is when someone thinks about the problem and then decides to do something. What's their thinking process here, what tool or process do I not know about that makes me confused?

Another way to see it is as a continuous generalization of the idea that you need n bits to represent 2^n equally likely alternatives.

I'm stuck on why we care about surprisal as `log 1/p`.

If your distribution has N equally-likely values, `p(x) = 1/N`, and information/surprisal `I(x) = log(N)`. In base 2, this is how many bits are required to specify exactly WHICH of the N values you're talking about.

If `x` is not a single one of the N states but an event consisting of `n(x)` states, then `I(x) = log(N) - log(n(x))`, suggesting it takes _somewhat less information_ to specify this particular state, and it correctly gives 0 if `n(x) = N`, i.e. there's only one state.

Exactly what this "less information" means is vague, but you might think of it in terms of compression: if you compress some stream of data which is sampled from `X` with probability `p(x)`, you could use use the shortest codes (0, 10, 11, etc) for the most common values with some "stop word" to say when the end of a datum is reached. `I(x)` captures this sense in general, but it might only become literally true in the limit of a very large stream of data with a very large dictionary.

I think the parent is also asking why we would expect surprisal to be additive rather than multiplicative like probabilities.

Times a constant!

Hartley was (as far as I know) the first person to recognise the usefulness of log probabilities in the context of measuring information [2]. It's a really amazing paper, for several reasons ... but one thing that really strikes me about it is how it's aged: the first half has an essentially timeless presentation of the essence of information, and the second has a now-quite-irrelevant presentation of how to make a better TV. I guess that was the really interesting problem at the time!

[1] https://people.math.harvard.edu/~ctm/home/text/others/shanno...

[2] http://dotrose.com/etext/90_Miscellaneous/transmission_of_in...

Good question. It's because -log P(X) is the number of bits of information I would need to eliminate any surprise about the value of X.

For example, let's say X is the value of an 8-bit register, and each of the 2^8=256 possible values has a probability of 1/256. I would need to learn all 8 bits to know the value of X: so Surprisal(X=x) = -log P(X=x) = -log 1/256 = 8.

Suppose I learned only that X is odd, which is equivalent to learning that the last bit of the register is equal to 1. The probability that X is 1 is 1/2, and so I would need to be provided with -log P(X is odd) = -log 1/2 = 1 bit of information to learn that X is odd.

Information can be thought of as simply the elimination of uncertainty. So Surprisal (uncertainty) is basically a measure of information, but instead of how much information you have, it's how much information you need to eliminate all uncertainty.