I maintain a reading list on Goodreads. I have a personal website with some blog posts, mostly technical stuff about math research. I am also on github, twitter, and mastodon.
Eigil Rischel
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[Question] Has anybody used quantification over utility functions to define “how good a model is”?
Where numbers come from
This is great!
An idea which has picked up some traction in some circles of pure mathematicians is that numbers should be viewed as the “shadow” of finite sets, which is a more fundamental notion.
You start with the notion of finite set, and functions between them. Then you “forget” the difference between two finite sets if you can match the elements up to each other (i.e if there exists a bijection). This seems to be vaguely related to your thing about being invariant under permutation—if a property of a subset of positions (i.e those positions that are sent to 1), is invariant under bijections (i.e permutations) of the set of positions, it can only depend on the size/number of the subset.
See e.g the first ~2 minutes of this lecture by Lars Hesselholt (after that it gets very technical)
My mom is a translator (mostly for novels), and as far as I know she exclusively translates into Danish (her native language). I think this is standard in the industry—it’s extremely hard to translate text in a way that feels natural in the target language, much harder than it is to tease out subtleties of meaning from the source language.
This post introduces a potentially very useful model, both for selecting problems to work on and for prioritizing personal development. This model could be called “The Pareto Frontier of Capability”. Simply put:
By an efficient markets-type argument, you shouldn’t expect to have any particularly good ways of achieving money/status/whatever - if there was an unusually good way of doing that, somebody else would already be exploiting it.
The exception to this is that if only a small amount of people can exploit an opportunity, you may have a shot. So you should try to acquire skills that only a small number of people have.
Since there are a lot of people in the world, it’s incredibly hard to become among the best in the world at any particular skill.
This means you should position yourself on the Pareto Frontier—you should seek out a combination of skills where nobody else is better than you at everything. Then you will have the advantage in problems where all these skills matter.
It might be important to contrast this with the economical term comparative advantage, which is often used informally in a similar context. But its meaning is different. If we are both excellent programmers, but you are also a great writer, while I suck at writing, I have a comparative advantage in programming. If we’re working on a project together where both writing and programming are relevant, it’s best if I do as much programming as possible while you handle as much as the writing as possible—even though you’re as good at me as programming, if someone has to take off time from programming to write, it should be you. This collaboration can make you more effective even though you’re better at everything than me (in the economics literature this is usually conceptualized in terms of nations trading with each other).
This is distinct from the Pareto optimality idea explored in this post. Pareto optimality matters when it’s important that the same person does both the writing and the programming. Maybe we’re writing a book to teach programming. Then even if I am actually better than you at programming, and Bob is much better than you at writing (but sucks at programming), you would probably be the best person for the job.
I think the Pareto frontier model is extremely useful, and I have used it to inform my own research strategy.
While rereading this post recently, I was reminded of a passage from Michael Nielsen’s Principles of Effective Research:
Say some new field opens up that combines field X and field Y. Researchers from each of these fields flock to the new field. My experience is that virtually none of the researchers in either field will systematically learn the other field in any sort of depth. The few who do put in this effort often achieve spectacular results.
I hadn’t, thanks!
I took the argument about the large-scale “stability” of matter from Jaynes (although I had to think a bit before I felt I understood it, so it’s also possible that I misunderstood it).
I think I basically agree with Eliezer here?
The Second Law of Thermodynamics is actually probabilistic in nature—if you ask about the probability of hot water spontaneously entering the “cold water and electricity” state, the probability does exist, it’s just very small. This doesn’t mean Liouville’s Theorem is violated with small probability; a theorem’s a theorem, after all. It means that if you’re in a great big phase space volume at the start, but you don’t know where, you may assess a tiny little probability of ending up in some particular phase space volume. So far as you know, with infinitesimal probability, this particular glass of hot water may be the kind that spontaneously transforms itself to electrical current and ice cubes. (Neglecting, as usual, quantum effects.)
So the Second Law really is inherently Bayesian. When it comes to any real thermodynamic system, it’s a strictly lawful statement of your beliefs about the system, but only a probabilistic statement about the system itself.
The reason we can be sure that this probability is “infinitesimal” is that macrobehavior is deterministic. We can easily imagine toy systems where entropy shrinks with non-neglible probability (but, of course, still grows /in expectation/). Indeed, if the phase volume of the system is bounded, it will return arbitrarily close to its initial position given enough time, undoing the growth in entropy—the fact that these timescales are much longer than any we care about is an empirical property of the system, not a general consequence of the laws of physics.
To put it another way: if you put an ice cube in a glass of hot water, thermally insulated, it will melt—but after a very long time, the ice cube will coalesce out of the water again. It’s a general theorem that this must be less likely than the opposite—ice cubes melt more frequently than water “demelts” into hot water and ice, because ice cubes in hot water occupies less phase volume. But the ratio between these two can’t be established by this sort of general argument. To establish that water “demelting” is so rare that it may as well be impossible, you have to either look at the specific properties of the water system (high number of particles the difference in phase volume is huge), or make the sort of general argument I tried to sketch in the post.
This may be poorly explained. The point here is that
is supposed to be always well-defined. So each state has a definite next state (since X is finite, this means it will eventually cycle around).
Since is well-defined and bijective, each is for exactly one .
We’re summing over every , so each also appears on the list of s (by the previous point), and each also appears on the list of s (since it’s in )
E.g. suppose and when , and . Then is . But - these are the same number.
Demystifying the Second Law of Thermodynamics
But then shouldn’t there be a natural biextensional equivalence ? Suppose , and denote . Then the map is clear enough, it’s simply the quotient map. But there’s not a unique map - any section of the quotient map will do, and it doesn’t seem we can make this choice naturally.
I think maybe the subcategory of just “agent-extensional” frames is reflective, and then the subcategory of “environment-extensional” frames is coreflective. And there’s a canonical (i.e natural) zig-zag
Does the biextensional collapse satisfy a universal property? There doesn’t seem to be an obvious map either or (in each case one of the arrows is going the wrong way), but maybe there’s some other way to make it universal?
Right, that’s a good point.
What do you think about “cognitive biases as an edge”?
One story we can tell about the markets and coronavirus is this: It was not hard to come to the conclusion, by mid-to-late February, that a global COVID-19 pandemic was extremely likely, and that it was highly probable it would cause a massive catastrophe in the US. A few people managed to take this evidence seriously enough to trade on it, and made a killing, but the vast majority of the market simply didn’t predict this fairly predictable course of events. Why not? Because it didn’t feel like the sort of thing that could happen. It was too outlandish, too weird.
If we take this story seriously, it makes sense to look for other things like this—cases where the probability of something is being underestimated—because it seems too weird, because it’s too unpleasant to think about, or for some other reason.
For example, metaculus currently estimates something like a 15% chance that Trump loses the election and refuses to concede. I we take that probability seriously, and assume something like that is likely to lead to riots, civil unrest, uncertainty, etc, would it make sense to try and trade on that? On the assumption that this is not priced in, because the possibility of this sort of crisis is not something that the market knows how to take seriously?
What are some reputable activist short-sellers?
Where do you go to identify Robinhood bubbles? (Maybe other than “lurk r/wallstreetbets and inverse whatever they’re hyping”).
I guess this question is really a general question about where you go for information about the market, in a general sense. Is it just reading a lot of “market news” type sites?
Thank you very much!
I guess an argument of this type rules out a lot of reasonable-seeming inference rules—if a computable process can infer “too much” about universal statements from finite bits of evidence, you do this sort of Gödel argument and derive a contradiction. This makes a lot of sense, now that I think about it.
[Question] How much is known about the “inference rules” of logical induction?
There is also predictionbook, which seems to be a similar sort of thing.
Of course, there’s also metaculus, but that’s more of a collaborative prediction aggregator, not so much a personal tool for tracking your own predictions.
If anyone came across this comment in the future—the CFAR Participant Handbook is now online, which is more or less the answer to this question.
The Terra Ignota sci-fi series by Ada Palmer depicts a future world which is also driven by “slack transportation”. The mechanism, rather than portals, is a super-cheap global network of autonomous flying cars (I think they’re supposed to run on nuclear engines? The technical details are not really developed). It’s a pretty interesting series, although it doesn’t explore the practical implications so much as the political/sociological ones (and this is hardly the only thing driving the differences between the present world and the depicted future)
I think, rather than “category theory is about paths in graphs”, it would be more reasonable to say that category theory is about paths in graphs up to equivalence, and in particular about properties of paths which depend on their relations to other paths (more than on their relationship to the vertices)*. If your problem is most usefully conceptualized as a question about paths (finding the shortest path between two vertices, or counting paths, or something in that genre, you should definitely look to the graph theory literature instead)
* I realize this is totally incomprehensible, and doesn’t make the case that there are any interesting problems like this. I’m not trying to argue that category theory is useful, just clarifying that your intuition that it’s not useful for problems that look like these examples is right.
This seems prima facie unlikely. If you’re not worried about the risk of side effects from the “real” vaccine, why not just take it, too (since the efficacy of the homemade vaccine is far from certain)?. On the other hand, if you’re the sort of person who worries about the side effects of a vaccine that’s been through clinical trials, you’re probably not the type to brew something up in your kitchen based on a recipe that you got off the internet and snort it.