Eigil Rischel

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


Kelly *is* (just) about logarithmic utility

This argument doesn't work because limits don't commute with integrals (including expected values). (Since practical situations are finite, this just tells you that the limiting situation is not a good model).

To the extent that the experiment with infinite bets makes sense, it definitely has EV 0. We can equip the space with a probability measure corresponding to independent coinflips, then describe the payout using naive EV maximization as a function - it is on the point and everywhere else. The expected value/integral of this function is zero.

EDIT: To make the "limit" thing clear, we can describe the payout after bets using naive EV maximization as a function , which is if the first values are , and otherwise. Then , and (pointwise), but .

The corresponding functions corresponding to the EV using a Kelly strategy have for all , but

Kelly *is* (just) about logarithmic utility

The source of disagreement seems to be about how to compute the EV "in the limit of infinite bets". I.e given bets with a chance of winning each, where you triple your stake with each bet, the naive EV maximization strategy gives you a total expect value of , which is also the maximum achievable overall EV. Does this entail that the EV at infinite bets is ? No, because with probability one, you'll lose one of the bets and end up with zero money.

I don't find this argument for Kelly super convincing.

  • You can't actually bet an infinite number of times, and any finite bound on the number of bets, even if it's , immediately collapses back to the above situation where naive EV-maximization also maximizes the overall expected value. So this argument doesn't actually support using Kelly over naive EV maximization in real life.

  • There are tons of strategies other than Kelly which achieve the goal of infinite EV in the limit. Looking at EV in the limit doesn't give you a way of choosing between these. You can compare them over finite horizons and notice that Kelly gives you better EV than others here (maximal geometric growth rate).... but then we're back to the fact that over finite time horizons, naive EV does even better than any of those.

The slopes to common sense

I don't wanna clutter the comments too much, so I'll add this here: I assume there was supposed to be links to the various community discussions of Why We Sleep (hackernews, r/ssc, etc), but these are just plain text for me.

Making Vaccine

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.

Recognizing Numbers

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)

Great minds might not think alike

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.

Being the (Pareto) Best in the World

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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.

Demystifying the Second Law of Thermodynamics

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.

Demystifying the Second Law of Thermodynamics

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.

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