The problem I'm trying to understand is more of a meta/proof-theoretic one: why do some arithmetical claims have a proof only when passing through non-arithmetical language?
I agree this is an interesting question. Thanks for pointing me to the speed-up theorem. I didn't know about that one. :)
This sounds horribly inefficient; intuitively, it sounds like that any "natural" statement provable in PA should be provable using tools from this system, and not by encoding concepts from a different field.
Yeah, I don't share that intuition. It feels like if that was true, there would be no other fields and everyone would be using arithmetic for everything at all times. I guess your phrasing of "natural" is doing a lot of work here.
There are fundamental rules of the universe that I don't yet understand. And for some reason, one of them seems to spell out: "thou shall use complex-valued analysis to study the behavior of prime numbers".
I am not at all a number theory expert and I am not quite sure what shape of explanation you are looking for here. One possible explanation I have here though is that you might be missing the forest for all the trees. From my outsider perspective the connection is already obvious in your introduction: Prime numbers -> modular arithmetic ~= arithmetic on circles -> complex numbers
If you have a problem involving operations on circles throwing complex analysis at it seems like the type of thing you would want to throw at the problem. The arrow Prime numbers -> modular arithmetic seems actually more worthy of a good compressed explanation.
I just watched the Veritasiums video. The forest fire simulation really made self-organized criticality click for me, when the sand pile analogy absolutely hadn't (though to be fair I had only read the vague description in Introduction to Complex Systems, which is absolutely inadequate compared to just seeing a simulation).
If the thing was really symmetrical like the post describes it should definitely be exploitable by someone, not necessarily smart, but with unconventional preferences?
Interesting! Makes sense.
If there's a way to make this version work for non-naive updates that seems good, and my understanding is it's mostly about saying for each new line "given that the above has happened, what are the odds of this observation?"
Yes that's it. Yeah I am not trying to defend the probability version of bayes rule. When I was trying to explain bayes rule to my wordcel gf, I was also using the odds ratio.
This version though? This I think most people could remember.
By most people you mean most people hanging around the lesswrong community because they know programming? I agree, an explanation that uses language that the average programmer can understand seems like a good strategy of explaining Bayes rule given the rationality communities demographics (above average programmers).
Maybe this is a case of Writing A Thousand Roads To Rome where this version happened to click with me but it's fundamentally just as good as many other versions. I suspect this is a simpler formulation.
Was it the code or the example that helped? The code is mostly fine. I don't think it is any simpler than the explanations here, the notation just looks scarier.
Either someone needs to point out where this math is wrong, or I'm just going to use this version for myself and for explaining it to others
This version is correct for naive bayes, but naive bayes is in fact naive and can lead you arbitrarily astray. If you wanted a non-naive version you would write something like this in pseudopython:
for i, E IN enumerate(EVIDENCE):
YEP *= CHANCE OF E IF all(YEP, EVIDENCE[:i])
NOPE *= CHANCE OF E IF all(NOPE, EVIDENCE[:i])
I see the case for starting with the naive version though, so this is more of a minor thing.
I don't see a lot more going for the bear example except for it being about something dramatic, so more memorable. Feels like you should be able to do strictly better examples. See Zane's objections in the other comment.
I like this post a lot. It might explain why I feel like an expert at addition, but not on addition. I notice when I am struggling with things like this in math, I often start blaming my own intellect instead of trying to understand what is making this hard and if this is perhaps just bad design that is to blame. The second approach seems much more likely to solve the problem. Noticing that word problems are harder seems like a good thing to notice, especially if you want to become an expert at using a particular math tool. For example I don't think I currently really get exterior products and searching for relevant word problems might be a good way to practice. LLMs might be useful in creating problems I can't solve (although I found it astonishing a while ago when Sonnet 3.5 was not able to consistently create word problems for applying bayes rule (~50% were just wrong)).
Suppose the agent's utility function is concave, i.e. the agent prefers (50% double wealth, 50% lose everything) over (100% wealth stays the same).
I think you meant to write convex here.
1Nice work in keeping up your public journal.
1
From my notes for the book 0 to 1: "Competition is destructive and not a sign of value".
For your alpha, look for secrets (things you know or are confident in, but no one else is). Create something that is 10x better than any alternative. Don't start the next restaurant. You want to be the next monopoly.
Not the focus of that book, but personally, I would also like to create value, not only capture it. So I'd aim not to start the next Elsevier, Coca-Cola, or Facebook, even though they have great profit margins. There are good and bad monopolies.