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This is really cool!
The example of inferring from the independence of and reminds me of some techniques discussed in Elements of Causal Inference. They discuss a few different techniques for 2-variable causal inference.
One of them, which seems to be essentially analogous to this example, is that if are real-valued variables, then if the regression error (i.e for some constant ) is independent of , it's highly likely that is downstream of . It sounds like factored sets (or some extension to capture continuous-valued variables) might be the right general framework to accommodate this class of examples.
Thanks (to both of you), this was confusing for me as well.
At least one explanation for the fact that the Fall of Rome is the only period of decline on the graph could be this: data becomes more scarce the further back in history you go. This has the effect of smoothing the historical graph as you extrapolate between the few datapoints you have. Thus the overall positive trend can more easily mask any short-term period of decay.
Lsusr ran a survey here a little while ago, asking people for things that "almost nobody agrees with you on". There's a summary here
This argument proves that
This is definitely true, and this is an inescapable feature of any (compact) dynamical system. However, somewhat paradoxically, it's consistent with the statement that, conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy.
In your time-diagrams, this might look something like this:
I.e when you occasionally swing down into a somewhat low-entropy state, it's much more likely that you'll go back to high-entropy than that you'll go further down. So once you observe that you're not in the maxentropy state, it's more likely that you'll increase than that you'll decrease.
(It's impossible for half of the mid-entropy states to continue to low-entropy states, because there are much more than twice as many mid-entropy states as low-entropy states, and the dynamics are measure-preserving).
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
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.
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.
(John made a post, I'll just post this here so others can find it: https://www.lesswrong.com/posts/Dx9LoqsEh3gHNJMDk/fixing-the-good-regulator-theorem)
My impression from skimming a few AI ETFs is that they are more or less just generic technology ETFs with different branding and a few random stocks thrown in. So they're not catastrophically worse than the baseline "Google, Microsoft and Facebook" strategy you outlined, but I don't think they're better in any real way either.