In recent posts by Manoj Gopalkrishnan and Marc Harper we’ve seen how not just entropy but relative entropy—the entropy of a probability distribution relative to the equilibrium distribution—is a driving force in chemistry and evolution. Now Tobias Fritz and I have finally finished our paper on this subject:
Very roughly, we proved that any reasonable measure of the information you gain when you to update your assumptions about the world based on discovering what a system is really doing must be some constant times the relative entropy.
I’ve blogged about this here before:
• Relative Entropy (Part 1): how various structures important in probability theory arise naturally when you do linear algebra using only the nonnegative real numbers.
• Relative Entropy (Part 2): a category related to statistical inference, and how relative entropy defines a functor on this category.
• Relative Entropy (Part 3): statement of our main theorem, which characterizes relative entropy up to a constant multiple as the only functor with a few nice properties.
Now that the paper is done, we’re having a nice conversation about it on the n-Category Café. Since I don’t want to splinter the conversation, I won’t enable comments here—please go there and join the fun!
But having blogged about this thrice before, what’s new?
One thing is that our conversation is getting more deeply into the category-theoretic aspects. Read the long parenthetical remarks in my post on the n-Café to get up to speed on that aspect.
Another is that by looking at our paper, you can actually see the proof of our result. As I mention on the n-Café.
The proof is surprisingly hard. Or maybe we’re just surprisingly bad at proving things. But the interesting thing is this: the proof is swift and effective in the ‘generic’ case—the case where the support of the probability measures involved is the whole set they’re living on, and the constant is finite.
It takes some more work to handle the case where the probability measures have smaller support.
But the really hard work starts when we handle the case that, in the end, has Then the proof becomes more like analysis than what you normally expect in category theory. We slowly corner the result, blocking off all avenues of escape. Then we close in, grab its neck, and strangle it, crushing its larynx ever tighter, as it loses the will to fight back and finally expires… still twitching.
We haven’t gotten into discussing this much yet, perhaps because the mathematicians on the n-Café are too dainty and civilized. But someone into analysis might be able to find a more efficient proof.
That would make me a bit sad—since why didn’t we find it?—but mainly happy—since this subject deserves to be clean and elegant. We really need a category-theoretic formulation of the second law of thermodynamics that’s suitable for studying complex networks: that’s the long-term goal here.