transhumanist_atom_understander

Karma: 383

transhumanist_atom_understander 4 Sep 2025 16:29 UTC
1 point
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in reply to: Gordon Seidoh Worley’s comment on: All Exponentials are Eventually S-Curves
I’m surprised at how hard it is for me to think of counterexamples.

I thought surely whale populations due to the slow generation time, but it looks like humpback whale populations have already recovered from whaling, and blue whales will get there before long.

Thinking again—in my baseball example, gravity is pulling the ball into the domain of applicability of the constant acceleration model.

Maybe what’s special about the exponential growth model is it implies escape from its own domain of applicability, in time that grows slowly (logarithmically) with the threshold.

transhumanist_atom_understander 4 Sep 2025 2:22 UTC
1 point
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on: Löb’s Lemma: an easier approach to Löb’s Theorem
I remember this by analogy to Curry’s paradox.

Where the sentence from Curry’s paradox says “If this statement is true, then $p$ ”, $Ψ$ says “if this statement is provable, then $p$ ”, that is, $□ Ψ \to p$ .

In Curry’s paradox, if the sentence is true, that would indeed imply that $p$ is true. And with $Ψ$ , the situation is analogous, but with truth replaced by provability: if $Ψ$ is provable, then $p$ is provable. That is, $□ Ψ \to □ p$ .

But, unlike in Curry’s paradox, this is not what $Ψ$ itself says! Replacing truth with provability has attenuated the sentence, destroyed its ability to cause paradox.

If only $□ p \to p$ , then we would have our paradox back… and that’s Löb’s theorem.

This is all about $□ Ψ \to □ p$ , just about one direction of the biimplication, whereas the post proves not just that but the other direction. It seems that only this forward direction is used in the proof at the end of the post though.

transhumanist_atom_understander 4 Sep 2025 1:49 UTC
4 points
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on: All Exponentials are Eventually S-Curves
You say “if we are to accurately model the world”...

If I am modelling the path of a baseball, and I write “F = mg”, would you “correct” me that it’s actually inverse square, that the Earth’s gravitation cannot stay at this strength to arbitrary heights? If you did, I would remind you that we are talking about a baseball game, and not shooting it into orbit—or conclude that you had an agenda other than determining where the ball lands.

What if I’m sampling from a population, and you catch me multiplying probabilities together, as if my draws are independent, as if the population is infinite? Yes there is an end to the population, but as long as it’s far away, the dependence induced by sampling without replacement is negligible.

Well, that’s the question, whether to include an effect in the model or whether it’s negligible. An effect like finite population size, diminishing gravity, or the “crowding” effects that turn an exponential growth model logistic.

And the question cannot be escaped just by noting the effect is important eventually.

transhumanist_atom_understander 3 Sep 2025 3:38 UTC
2 points
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on: Yudkowsky on “Don’t use p(doom)”
Eliezer in 2008, in When (Not) To Use Probabilities, wrote:

To be specific, I would advise, in most cases, against using non-numerical procedures to create what appear to be numerical probabilities. Numbers should come from numbers.

transhumanist_atom_understander 2 Sep 2025 13:50 UTC
1 point
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in reply to: dr_s’s comment on: A quantum equivalent to Bayes’ rule
Yeah… well, I thought of the $Z$ because it sounds like we’re getting the probabilities of $Y$ from some experiment. So $Z = z$ is the results of the experiment, which in this case is a vector of frequencies. When I put it like that, it sounds like it’s is just a rhetorical device for saying that we have given probabilities of $Y$ .

But I still seem to need $Z$ for my dictionary. I have $γ (x) = P [X = x]$ . What is $γ^{'} (x)$ ? It is some kind of updated probability of $X = x$ , right? Like we went from one probability to the other by doing an experiment. If I didn’t write $γ^{'} (x) = P [X | Z = z]$ , I’d need something like $γ (x) = P_{1} [X = x]$ and $γ^{'} (x) = P_{2} [X = x]$ .

Reading again, it seems like this is exactly Jeffrey conditionalization. So whether you include some extra variable just depends on what you think of Jeffrey conditionalization.

I feel like I’m missing something, though, about what this experiment is and means. For example, I’m not totally clear on whether we have one state $X$ , and a collection of replicates of state $Y$ ; or is it a collection of replicates of $(X, Y)$ pairs?

Looking at the paper, I see the connection to Jeffrey conditionalization is made explicitly. And it mentions Pearl’s “virtual evidence method”; is this what he calls introducing this $Z$ ? But no clarity on exactly what this experiment is. It just says:

But how should the above be generalized to the situation where the new information does not come in the form of a definite value $y_{0}$ for $Y$ , but as “soft evidence,” i.e., a probability distribution $τ (y)$ ?”

By the way, regarding your coin toss example, I can at least say how this is handled in Bayesian statistics. There are separate random variables for each coin toss. $Y_{1}$ is the first, $Y_{2}$ is the second, etc. If you have $n$ coin tosses, then your sample is a vector $\to Y$ containing $Y_{1}$ to $Y_{n}$ . Then the posterior probability is $P [loaded | \to Y = \to y]$ . This will be covered in any Bayesian statistics textbook as “the Bernoulli model”. My class used Hoff’s book, which provides a quick start.

I guess this example suggests a single unknown $X$ (whether the coin is loaded or not) and replicates of $Y$ .

transhumanist_atom_understander 2 Sep 2025 2:05 UTC
8 points
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on: A quantum equivalent to Bayes’ rule
The “Classical derivation” made more sense to me after translating it to standard probability notation, so I’m commenting to share the “dictionary” I made for it, as well as the unexpected extra assumption I had to make.

The obvious:

$γ (x) = P [X = x]$

$φ (y | x) = P [Y = y | X = x]$

$^φ (x | y) = P [X = x | Y = y]$

It got tricky with $τ$ . Instead of observing $Y = y$ , we observe something else that gives us a probability distribution over $Y$ . I considered this “something else” to be the value of some other unknown: $Z = z$ . The probability distribution over $y$ is a conditional distribution:

$τ (y) = P [Y = y | Z = z]$

Hate to have $z$ on only one side like that… maybe I should have called it $τ_{z}$ … but I’ll leave it as is.

Then,

$γ^{'} (x) = \sum j P [X = x | Y = y_{j}] P [Y = y_{j} | Z = z]$

Not quite the right formula for a simple interpretation of $γ^{'}$ … if only

$P [X = x | Y = y_{j}] = P [X = x | Y = y_{j}, Z = z]$

This is conditional independence, which could be represented with this Bayes net:

$Z \to Y \to X$

Then, we have

$γ^{'} (x) = P [X = x | Z = z]$

That completes the dictionary.

So to do what feels like ordinary probability theory, I had to introduce this extra unknown $Z$ so that we have something to observe, and then to assume that $Z$ only provides information about $Y$ (and indirectly about $X$ , through $Y$ ).

The way you described $τ$ as some probability distribution resulting from an observation, but not a conditional distribution, is in philosophy called Jeffrey conditionalization. The Stanford Encyclopedia of Philosophy gives this example:

A gambler is very confident that a certain racehorse, called Mudrunner, performs exceptionally well on muddy courses. A look at the extremely cloudy sky has an immediate effect on this gambler’s opinion: an increase in her credence in the proposition (muddy) that the course will be muddy—an increase without reaching certainty. Then this gambler raises her credence in the hypothesis (win) that Mudrunner will win the race, but nothing becomes fully certain. (Jeffrey 1965 [1983: sec. 11.3])

The idea is, we go from one probability distribution over ${muddy, \neg muddy}$ to another, without becoming certain of anything. My introduction of $Z$ corresponds to introducing an unknown representing the status of the sky. I would say we are conditioning on $Z = cloudy$ .

I recalled vaguely that Jaynes discussed Jeffrey conditionalization in Probability Theory, and criticized it for holding only in a special case. I took a look, and sure enough, it’s in section 5.6, and he’s pointing out exactly what I did, right down to the arrows, though he calls it a “logic flow diagram” rather than identifying it as a Pearl-style Bayes net.

transhumanist_atom_understander 25 Apr 2025 13:18 UTC
1 point
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on: Unknown Probabilities
The last formula in this post, the conservation of expected evidence, had a mistake which I’ve only just now fixed. Since I guess it’s not obvious even to me, I’ll put a reminder for myself here, which may not be useful to others. Really I’m just “translating” from the “law of iterated expectations” I learned in my stats theory class, which was:

$E [E [X | Y]] = E [X]$

This is using a notation which is pretty standard for defining conditional expectations. To define it you can first consider the expected value given a particular value of the random variable $Y$ . Think of that as a function of that particular value: $f (y) = E [X | Y = y]$ Then we define conditional expectation as a random variable, obtained from plugging in the random value of $Y$ : $E [X | Y] = f (Y)$ The problem with this notation is it gets confusing which capital letters are random variables and which are propositions, so I’ve bolded random variables. But it makes it very easy to state the law of iterated expectations.

The law of iterated expectations also holds when “relativized”. That is, $E [E [X | Y] | B] = E [X | B]$ where $B$ is an event. If we wanted to stick to just putting random variables behind the conditional bar we could have used the indicator function of that event.

And this translates to the statement in my post. $X$ is an indicator for the event $H$ , which makes a conditional expectation of it a conditional probability of $H$ . So $E [X | Y]$ is $Θ$ . Our proposition $B$ is the background information $B$ , I used the same symbol there. And the right hand side is another expectation of an indicator and therefore also a probability.

I really didn’t want to define this notation in the post itself, but it’s how I’m trained to think of this stuff, so for my own confidence in the final formula I had to write it out this way.

transhumanist_atom_understander 7 Apr 2025 1:52 UTC
1 point
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on: How Gay is the Vatican?
It would be nice if you had the sexes of the siblings, since it’s supposedly only the older brothers that count, though I don’t really expect that to change anything.

Really the important thing is just to separate birth order from family size. Usually the way I think of this is, we can look at number of older brothers, with a given number of older siblings. I like this setup because it looks like a randomized trial. I have two older siblings, so do you, meiosis randomizes their sexes.

But I guess with the data you have you can look at birth order with a given family size, so we don’t have to worry about the effect of a larger or smaller family. I… don’t think this is what you did? Did I misunderstand something? It seems like if cardinals come from smaller families, that would show up as lower birth orders.

With 9 million people I’d just split it into categories by number of siblings, with your data I’m not sure.

transhumanist_atom_understander 18 Feb 2025 20:53 UTC
3 points
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on: AI #103: Show Me the Money
After reading this post, I handed over $200 for a month of ChatGPT pro and I don’t think I can go back. o1-pro and Deep Research are next level. o1-pro often understands my code or what I’m asking about without a whole conversation of clarifying, whereas other models it’s more work than it’s worth to get them focused on the real issue rather than something superficially similar. And then I can use “Deep Research” to get links to webpages relevant to what I’m working on. It’s like… smart Google, basically. Google that knows what I’m looking for. I never would have known this existed if I didn’t hand over the $200.

transhumanist_atom_understander 5 Feb 2025 1:55 UTC
1 point
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in reply to: Alexander Gietelink Oldenziel’s comment on: Alexander Gietelink Oldenziel’s Shortform
Depends entirely on Cybercab. A driverless car can be made cheaper for a variety of reasons. If the self-driving tech actually works, and if it’s widely legal, and if Tesla can mass produce it at a low price, then they can justify that valuation. Cybercab is a potential solution to the problem that they need to introduce a low priced car to get their sales growing again but cheap electric cars is a competitive market now without much profit margin. But there’s a lot of ifs.

transhumanist_atom_understander 3 Feb 2025 18:46 UTC
3 points
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in reply to: Steven Byrnes’s comment on: The Self-Reference Trap in Mathematics
Yeah, just went through this whole same line of evasion. Alright, the Collatz conjecture will never be “proved” in this restrictive sense—and neither will the Steve conjecture or the irrationality of √2—do we care? It may still be proved according to the ordinary meaning.

transhumanist_atom_understander 3 Feb 2025 18:09 UTC
1 point
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on: What are the good rationality films?
The pilot episode of NUMB3RS.

The tie-in to rationality is that instead of coming up with a hypothesis about the culprit, the hero comes up with algorithms for finding the culprit, and quantifies how well they would have worked applied to past cases.

It’s really a TV episode about computational statistical inference, rather than a movie about rationality, but it’s good media for cognitive algorithm appreciators.

transhumanist_atom_understander 3 Feb 2025 18:01 UTC
1 point
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in reply to: Alister Munday’s comment on: The Self-Reference Trap in Mathematics
Alright, so Collatz will be proved, and the proof will not be done by “staying in arithmetic”. Just as the proof that there do not exist numbers p and q that satisfy the equation p² = 2 q² (or equivalently, that all numbers do not satisfy it) is not done by “staying in arithmetic”. It doesn’t matter.
What links here?
- transhumanist_atom_understander's comment on The Self-Reference Trap in Mathematics by Alister Munday (3 Feb 2025 18:46 UTC; 3 points)

transhumanist_atom_understander 3 Feb 2025 17:47 UTC
1 point
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in reply to: Alister Munday’s comment on: The Self-Reference Trap in Mathematics
Not off the top of my head, but since a proof of Collatz does not require working under these constraints, I don’t think the distinction has any important implications.

transhumanist_atom_understander 3 Feb 2025 17:43 UTC
1 point
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in reply to: Alister Munday’s comment on: The Self-Reference Trap in Mathematics
We can eliminate the concept of rational numbers by framing it as the proof that there are no integer solutions to p² = 2 q²… but… no proof by contradiction? If escape from self-reference is that easy, then surely it is possible to prove the Collatz conjecture. Someone just needs to prove that the existence of any cycle beyond the familiar one implies a contradiction.

transhumanist_atom_understander 3 Feb 2025 17:34 UTC
1 point
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in reply to: Alister Munday’s comment on: The Self-Reference Trap in Mathematics
Yes, the proof that there’s no rational solution to x²=2. It doesn’t require real numbers.

transhumanist_atom_understander 3 Feb 2025 17:19 UTC
1 point
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in reply to: Alister Munday’s comment on: The Self-Reference Trap in Mathematics
Yeah, I think this is the distinction I’m struggling with. To start with proofs in Euclid, why can I prove that the square root of two is irrational? Why can I prove there are infinite prime numbers? If I saw that they are escaping this self-reference somehow, maybe I’d get the point. And without that, I don’t see that I can rule out such an escape in Collatz.

2024 was the year of the big battery, and what that means for solar power

transhumanist_atom_understander1 Feb 2025 6:27 UTC

36 points

1 comment8 min readLW link

transhumanist_atom_understander 13 Jan 2025 20:33 UTC
6 points
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on: Chance is in the Map, not the Territory
Yes, de Finetti’s theorem shows that if our beliefs are unchanged by exchanging members of the sequence, that’s mathematically equivalent to having some “unknown probability” that we can learn by observing the sequence.

Importantly, this is always against some particular background state of knowledge, in which our beliefs are exchangeable. We ordinary have exchangeable beliefs about coin flips, for example, but may not if we had less information (such as not knowing they’re coin flips) or more (like information about the initial conditions of each flip).

In my post on unknown probabilities, I give more detail on how they are definedc which turns out to involve a specific state of background knowledge, so they only act like a “true” probability relative to that background knowledge. And how they can be interpreted as part of a physical understanding of the situation.

Personally, rather than observing that my beliefs are exchangeable and inferring an unknown probability as a mathematical fiction, I would rather “see” the unknown probability directly in my understanding of the situation, as described in my post.

transhumanist_atom_understander 11 Jan 2025 3:23 UTC
1 point
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in reply to: Dmitry Vaintrob’s comment on: The Laws of Large Numbers
Well, usually I’m not inherently interested in a probability density function, I’m using it to calculate something else, like a moment or an entropy or something. But I guess I’ll see what you use it for in future posts.

transhumanist_atom_understander

2024 was the year of the big bat­tery, and what that means for so­lar power

2024 was the year of the big battery, and what that means for solar power