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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.
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.)
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)
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.
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.
At my local Barnes and Nobles, I cannot access slatestarcodex.com nor putanumonit.com. Have never had any issues accessing any other websites (not that I've tried to access genuinely sketchy websites there). The wifi there is titled Bartleby, likely related to Bartleby.com, whereas many other Barnes and Nobles have wifi titled something like "BNWifi". I have not tried to access these websites at other Barnes yet.
The hot hand fallacy: seeing data that is typical for independent coin flips as evidence for correlation between adjacent flips.
The hot hand fallacy fallacy (Miller, Sanjurjo 2018): Not correcting for the fact that amongst random length-k (k>2) sequences of independent coin tosses with at least one heads before toss k, the expected proportion of (heads after heads)/(tosses after heads) is less than 1/2.
The hot hand fallacy fallacy fallacy: Misinterpreting the above observation as a claim that under some weird conditioning, the probability of Heads given you have just seen Heads is less than 1/2 for independent coin tosses.
Has to be a python code; allowing arbitrary non-computable natural language descriptions gets hairy fast
Random thought after reading "A model of UDT with a halting oracle": imagine there are two super-intelligent AIs A and B, suitably modified to have access to their own and each other's source codes. They are both competing to submit a python code of length at most N which prints the larger number, then halts (where N is orders of magnitude larger than the code lengths of A and B). A can try to "cheat" by submitting something like exec(run B on the query "submit a code of length N that prints a large number, then halts") then print(0), but B can do this as well. Supposing they must submit to a halting oracle that will punish any AI that submits a non-halting program, what might A and B do?
The intended question, I think, is if you were to find a dictionary for some alien language (not a translators dictionary, but a dictionary for people who speak that language to look up definitions of words), can you translate most of the dictionary to English? What if you additionally had access to large amounts of conversations in that language, without any indication of what the aliens were looking at/doing at the time of the conversation?
Predictive Clustering: Whenever your writing is predictable (for example, when responding to something or after the first few sentences of a new post), an LLM could vaguely predict the points you might make. It could cluster these points, allowing you to point and click on the relevant cluster. For instance, in a political piece, you might first click, "I [Agree | Disagree | Raise Interesting Other Point | Joke]." You then select "Raise interesting point," and it presents you with 5-20 points you might want to raise, along with a text box to add your own. Once you add your point, you can choose a length.
This seems like something that is very likely to come into existence in the near future, but I hope does not. Not only does it rob people of the incredibly useful practice of crafting their own arguments, I think putting better words in the user's mouth than the user planned to say can influence the way the user actually thinks.
Frequentist and Bayesian reasoning are two ways to handle Knightian uncertainty. Frequentism gives you statements that are outright true in the face of this uncertainty, which is fantastic. But this sets an incredibly high bar that is very difficult to work with.
For a classic example, let's say you want have a possibly biased coin in front of you and you want to say something about its rate of heads. From frequentism, you can lock in a method of obtaining a confidence interval after, say, 100 flips and say "I'm about to flip this coin 100 times and give you a confidence interval for p_heads. The chance that the interval will contain p_heads is at least 99%, regardless of what the true value of p_heads is" There's no Bayesian analogue.
Now let's say I had a complex network of conditional probability distributions with a bunch of parameters which have Knightian uncertainty. Getting confidence regions will be extremely expensive, and they'll probably be way too huge to be useful. So we put on a convenient prior and go.
ETA: Randomized complexity classes also feel fundamentally frequentist.
Not fully, unfortunately. Although a baseline would be asking an LLM to convert my latex file into markdown that allows mathjax
Someone recently tried to sell me on the Ontological Argument for God which begins with "God is that for which nothing greater can be conceived." For the reasons you described, this is completely nonsensical, but it was taken seriously for a long time (even by Bertrand Russell!), which made me realize how much I took modern logic for granted
I didn't think much of your comment at the time, but I think it's extremely central to the whole thing now. We go from unconscious to conscious almost all at once.
Are there any other nice decision problems that are low? A quick search only reveals existence theorems.
Intuitive guess: Can we get some hierarchy from oracles to increasingly sparse subsets of the digits of Chaitin's constant?
*The last two bullet points. Meta-consciousness and self-consciousness
I meant if you had any suggested rewords, because there don't seem to be any perfect definitions of these concepts.
"Easy problems of consciousness" is an established term that is a bit better-defined than consciousness. By transcending, I just meant beyond what can be explained by solving the easy problems of consciousness
This was actually what I meant by a version of panpsychism that seemed to be the natural conclusion of humans having subjective experiences, but a conclusion I want to see if I can avoid.
I tried some different definitions of consciousness while writing this point, until settling on "able have subjective experiences that transcend the 'easy problems of consciousness'"
Do you have any suggestions for making this more precise?
I'd like to explore these in more depth, but for now I'll just reduce all the angles you provided to the helpful summaries/applications you provided. I'll call the perspective of going from adult human to zygote the "physical history" and the perspective of going up the ancestral tree as the "information history" (for simplicity, maybe we stop as soon as we hit a single-celled organism).
- Sentience: This feels like a continuous thing that gets less and less sophisticated as we go up the information history. In each generation, the code gets a little better at using the laws of physics and chemistry to preserve itself. Of course if one has a threshold for what counts as sentience, it will cross it at some point, but this still strikes me as continuous.
- Wakefulness: This would strike me as a quantized thing from both the information and physical history perspective. At some point in both histories, the organism/cell would pick up some cyclic behavior.
- Intentionality: I'd need to look more at this, because my interpretation of your first sentence doesn't make sense with the second.
- Phenomonal, Self-Consciousness, Meta-Consciousness: Definitely quantized in both perspectives
When I was thinking of subjective experience, I think the only concepts here that are either weaker or stronger than what I had in mind are the last two. For the rest, I think I can both imagine a robot that satisfies the conditions and imagine a conscious being that does not satisfy the condition.
But the last two still feel too strong. I will think more about it.
That's a bit of a long read, and both your endorsement and the title seem too strong to be believable. If a few more people endorse that it's worth reading, I'll give it a go!
Very nice! Notice that if you write as , and play around with binomial coefficients a bit, we can rewrite this as:
which holds for as well, in which case it becomes the derivative product rule. This also matches the formal power series expansion of , which one can motivate directly
(By the way, how do you spoiler tag?)
This is true, but I'm looking for an explicit, non-recursive formula that needs to handle the general case of the kth anti-derivative (instead of just the first).
The solution involves doing something funny with formal power series, like in this post.
Here's a puzzle I came up with in undergrad, based on this idea:
Let be a function with nice derivatives and anti-derivatives (like exponentials, sine, or cosine) and be a polynomial. Express the th anti-derivative of in terms of derivatives and anti-derivatives of and .
Can provide link to a post on r/mathriddles with the answer in the comments upon request
Suppose we don't have any prior information about the dataset, only our observations. Is any metric more accurate than assuming our dataset is the exact distribution and calculating mutual information? Kind of like bootstrapping.
For the second paragraph, we're assuming this AI has not made a mistake in predicting human behavior yet after many, many trials in different scenarios. No exact probability. We're also assuming perfect levels of observation, so we know that they pressed a button, bombs are heading over, and any observable context behind the decision (like false information).
The first paragraph contains an idea I hadn't considered, and it might be central to the whole thing. I'll ponder it more.
I didn't get around to providing more clarity. I'll do that now:
- Both parties would click the button if it was clear that the other party would not click the button in retaliation. This way they do not have to worry about being wiped off the map.
- The two parties would both prefer a world in which only the other party survives to a world without any humanity.
We know that the other party will click the button if and only if they predict with extremely high confidence that we will not retaliate. Our position is the same.
It's extremely beautiful, and seems like it would serve as a nice introduction to the website that isn't subject to the same random noise as the front page.
I really like 'leastwrong' in the url and top banner (header?), but I could see how making 'The LeastWrong' the actual title could rub off on some as pretentious.
Thanks for your answer; this explains why I was not able to find any related discussion on this. I read this article recently: https://www.lesswrong.com/posts/c3wWnvgzdbRhNnNbQ/timeless-decision-theory-problems-i-can-t-solve and misremembered it as a defense of evidential decision theory, instead of a different decision theory altogether.
So from a timeless decision theory perspective, is it correct to say that one would press the button? And from both EDT/CDT perspectives, one would not press the button (assuming they value the other country staying alive over no country staying alive)?
I'm not sure if these kind of comments are acceptable on this site, but I just wanted to say thank you for this sequence. I doubt I will significantly change my life after reading this, but I hope to change it at least a little in this direction.
Viewing myself as a reinforcement learning agent that balances policy improvement (taking my present model and thinking about how to tweak my actions to optimize rewards assuming my model is correct) and exploration (observing how the world actually responds to certain actions to update the model), I have historically spent far too much time on policy improvement.
This sequence provides a nice set of guidelines and methods to pivot gears and really think about what it even means to improve ones model of the world, in a way that seems... fun? fulfilling? I hope to report back on this in a few months and say how it's gone; there is a high probability that I fall back into old habits, but I hope I do not.
In case it hasn't crossed your mind, I personally think it's helpful to start in the setting of estimating the true mean of a data stream. A very natural choice estimator for is the sample mean of the which I'll denote . This can equivalently be formulated as the minimizer of .
Others have mentioned the normal distribution, but this feels secondary to me. Here's why - let's say , where is a known continuous probability distribution with mean 0 and variance 1, and are unknown. So the distribution of each has mean and variance (and assume independence).
What must be for the sample mean to be the maximum likelihood estimator of ? Gauss proved that it must be , and intuitively it's not hard to see why it would have to be of the form .
So from this perspective, MSE is a generalization of taking the sample mean, and asking the linear model to have gaussian errors is necessary to formally justify MSE through MLE.
Replace sample mean with sample median and you get the mean absolute error.
Is there no way to salvage it via a Nash bargaining argument if the odds are different? Or at least, deal with scenarios where you have x:1 and 0:1 odds (i.e. you can only bet on heads)?