Imo if you could really choose a point uniformly at random in [0,1], then things like Vitali sets philosophically shouldn’t exist (but I’ve gotten attacked on reddit for this reasoning, and I kinda don’t want to get into it). But this is why probability theory is phrased in terms of sigma algebras and whatnot to model what might happen if we really could choose uniformly at random in [0,1] instead of directly referring to such a platonic process. One could get away with being informal in probability theory by referring to such a process (and imo one should for the sake of grasping theorems), but then you have issues with the axiom of choice, as you mentioned. (I don’t think any results in probability theory invoke a version of the axiom of choice strong enough to construct non-measurable sets anyway, but I could be wrong.)
notfnofn
Also as additional theorems about a given category arise, and various equivalencies are proven, one often ends up with definitions that are much “neater” than the original. But there is sometimes value in learning the historical definitions.
No, but it’s exactly what I was looking for, and surprisingly concise. I’ll see if I believe the inferences from the math involved when I take the time to go through it!
We could also view computation through the lens of Turing Machines, but then that raises the argument of “what about all these quantum shenanigans, those are not computable by a turing machine”.
I enjoyed reading your comment, but just wanted to point out that a quantum algorithm can be implemented by a classical computer, just with a possibly exponential slow down. The thing that breaks down is that any O(f(n)) algorithm on any classical computer is at worst O(f(n)^2) on a Turing machine; for quantum algorithms on quantum computers with f(n) runtime, the same decision problem can be decided in (I think) O(2^{(f(n)}) runtime on a Turing machine
This pacifies my apprehension in (3) somewhat, although I fear that politicians are (probably intentionally) stupid when it comes to interpreting data for the sake of pushing policies
To add: this seems like the kind of interesting game theory problem I would expect to see some serious work on from members in this community. If there is such a paper, I’d like to see it!
Currently trying to understand why the LW community is largely pro-prediction markets.
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Institutions and smart people with a lot of cash will invest money in what they think is undervalued, not necessarily in what they think is the best outcome. But now suddenly they have a huge interest in the “bad” outcome coming to pass.
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To avoid (1), you would need to prevent people and institutions from investing large amounts of cash into prediction markets. But then EMH really can’t be assumed to hold
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I’ve seen discussion of conditional prediction markets (if we do X then Y will happen). If a bad foreign actor can influence policy by making a large “bad investment” in such a market, such that they reap more rewards from the policy, they will likely do so. A necessary (but I’m not convinced sufficient) condition for this is to have a lot of money in these markets. But then see (1)
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The pivotal time in my life where I finally broke out of my executive dysfunction and brain fog involved going to an area on campus that was completely abandoned over the summer with no technology, just a paper and pencil and a math book I was trying to get through while my wife was working on her experiments a building away (with my phone).
There wasn’t even a clock there.
The first few days, I did a little work then slept (despite not being slee-deprived). Then I started adding some periodic exercise. Then I started bringing some self-help books and spent some time reading those as well. Eventually, I stopped napping and spent the whole time working, reading, or exercising.
It’s not like I never went back to being unproductive for stretches of time after that summer, but I was never as bad as I was before that.
Not trying to split hairs here, but here’s what was throwing me off (and still is):
Let’s say I have an isomorphism: sequential states of a brain molecules of a rock
I now create an encoding procedure: physical things txt file
Now via your procedure, I consider all programs which map txt files to txt files such that
and obtain some discounted entropy. But isn’t doing a lot of work here? Is there a way to avoid infinite regress?
It feels like this a semantic issue. For instance, if you asked me if Euclid’s algorithm produces the gcd, I wouldn’t think the answer is “no, until it runs”. Mathematically, we often view functions as the set of all pairs (input,output), even when the input size is infinite. Can you clarify?
While I sort of get what you’re going for (easy interpretability of the isomorphism?), I don’t really a see a way to make this precise.
I’m having a little trouble understanding how to extend this toy example. You meant for these questions to all be answered “yes”, correct?
I think any operational definition of subjective experience would vacuously be preserved by an isomorphism, by definition of an isomorphism. But if your mind ever gets uploaded, you see/remember this conversation, and you feel that you are self-aware in any capacity, that would be a falsification of the claim that mind uploads don’t have subjective experience.
Would your intuition suggest that a computation by hand produces the same kind of experience as your brain? Your intuition reminds me of the strange mathematical philosophy of ultrafinitism, where even mathematical statements that require a finite amount of computation to verify do not have a truth value until they are computed.
[Question] Isomorphisms don’t preserve subjective experience… right?
I was thinking about this a few weeks ago. The answer is your units are related to the probability measure, and care is needed. Here’s the context:
Let’s say I’m in the standard set-up for linear regression: I have a bunch of input vectors and for some unknown and the outputs are independent with distributions
Let denote the matrix whose th row is , assumed to be full rank. Let denote the random vector corresponding to the fitted estimate of using ordinary least squares linear regression and let denote the sum of squared residuals. It can be shown geometrically that:
(informally, the density of is that of the random variable corresponding to sampling a multivariate gaussian with mean and covariance matrix , then sampling an independent distribution and dividing by the result). A naive undergrad might misinterpret this as meaning that after observing and computing :
But of course, this can’t be true in general because we did not even mention a prior. But on the other hand, this is exactly the family of conjugate priors/posteriors in Bayesian linear regression… so what possibly-improper prior makes this the posterior?
I won’t spoil the whole thing for you (partly because I’ve accidentally spent too much time writing this comment!) but start with just and and:
Calculate the exact posterior density of desired in terms of
Use Bayes theorem to figure out the prior
I personally messed up several times on step 2 because I was being extremely naive about the “units” cancelling in Bayes theorem. When I finally made it all precise using measures, things actually cancelled properly and got the correct improper prior distribution on .
(If anyone wants me to finish fleshing out the idea, please let me know).
Reading “Thinking Fast and Slow” for the first time, and came across an idea that sounds huge if true: that the amount of motivation one can exert in a day is limited. Given the replication crisis, I’m not sure how much credence I should give to this.
A corollary would be to make sure ones non-work daily routines are extremely low willpower when it’s important to accomplish a lot during the work day. This flies in the face of other conventional wisdom I’ve heard regarding discipline, even granting the possibility that the amount of total will-power one can exert over each day can increase with practice.
Anecdotally, my best work days typically start with a small amount of willpower (cold shower night before, waking up early, completing a very short exercise routine, prepping brunch, and biking instead of driving to a library/coffee shop). The people I know who were the best at puzzles and other high effort system-2 activities were the type who would usually complete assignments in school, but never submit their best work.
First, all of this is pretty standard knowledge in quantitative finance, and so there is little reason to believe this effect hasn’t yet been arbitraged away
I’m a little confused here: even if everyone was doing this strategy, would it not still be rational? Also what is your procedure for estimating the for real stocks? I ask because what I would naively guess (IRR) is probably an unbiased estimator for something like (not a finance person; just initial thoughts after reading).
I assume that it’s harder to have public bathrooms when you have a substantial homeless population. There’s a fear that they’ll do drugs in there or desecrate the place.
I was briefly part of an organization that tried to solve this problem by having a portable station for homeless people to use the bathroom, take a shower, brush, and change (they were also given inexpensive undergarments + cleaning equipment). While doing that, I never experienced any of the above issues but there was also an establishment of trust because the homeless people and the volunteers would interact regularly. I wonder if this can extrapolate.
In places without a homeless problem, I’ve never had an issue finding a place to use the bathroom without buying anything. I usually buy something after as a courtesy, but I never promise the storeowner or anything.
ETA: in upscale areas in the East Coast, I often can find public bathrooms, and they’re in good shape. I don’t travel too much, so I don’t have a whole lot of data points.
I’d be surprised if it could be salvaged using infinitesmals (imo the problem is deeper than the argument from countable additivity), but maybe it would help your intuition to think about how some Bayesian methods intersect with frequentist methods when working on a (degenerate) uniform prior over all the real numbers. I have a draft of such a post that I’ll make at some point, but you can think about univariate linear regression, the confidence regions that arise, and what prior would make those confidence regions credible regions.