1mo42

The mathematics of a latent variable model expresses the probabilities, p(x), for observations x as marginal probabilities integrating over unobserved z. That is, p(x) = integral over z of p(x,z), where p(x,z) is typically written as p(z)p(x|z).

It's certainly correct that nothing in this formulation says anything about whether z captures the "causes" of x.

However, I think it sometimes is usefully seen that way. Your presentation would be clearer if you started with one or more examples of what you see as typical models, in which you argue that ...

11mo

Hi Dr. Neal,
Wow, I studied your work in grad school! (And more recently your paper on Gaussian Processes). Quite an honor to get a comment from you. Just as an aside, I am not sure if my figure is visible, can you see it? I set it as the thumbnail, but I don't see it anywhere. In case it doesn't, it is here:
https://www.mlcrumbs.com/img/epistemology-of-representation.png
I think I need to change some labels, I realize now that I have been using 'x' ambiguously - sometimes as a model input, and sometimes to represent the bedrock physical system. But, to clarify, I'll use your vision example, but add temporality:
* zt: position, orientation and velocity of some object at time t
* xt: pixel values from a video camera
* ϕt: physical state (particles, forces) of the relevant slice of space at time t (includes the object and photons emanating from it which have hit the camera)
Actually your example of a typical vision model is one example where I'd argue this, though I fear you might think this is a trivial splitting of hairs.
In any case, I'll first assume you agree that "causation" by any use of the term, must be temporal - causes must come before effects. So, to a modified question: why wouldn't zt−δ be usefully seen as causing xt? In this case, let's assume then that delta is the time required for photons to travel from the object's surface to the camera.
What I'm more saying is that zt, or zt−δ are platonic, non-physical quantities. It is ϕt−δ which is causing ϕt, and xt is just a slice of ϕt. Or, if you like, xt could be seen as a platonic abstraction of it.
I would also add though that at best, zt−δ could at best be interpreted as an aspect of something that caused the pixels x. Of course, just position, orientation and velocity of an object aren't enough to determine colors of all pixels.
This vision example is one example in which the zt representations are very much rigid-body summaries, and so it seems useful to strongly identify them as "causes".

OK. My views now are not far from those of some time ago, expressed at https://glizen.com/radfordneal/res-bayes-ex.html

With regard to machine learning, for many problems of small to moderate size, some Bayesian methods, such as those based on neural networks or mixture models that I've worked on, are not just theoretically attractive, but also practically superior to the alternatives.

This is not the case for large-scale image or language models, for which any close approximation to true Bayesian inference is very difficult computationally.

However, I ...

OK. I think we may agree on the technical points. The issue may be with the use of the word "Bayesian".

Me: But they *aren't* guaranteed to eventually get a Bayesian to think the null hypothesis is likely to be false, when it is actually true.

You: Importantly, this is false! This statement is wrong if you have only one hypothesis rather than two.

I'm correct, by the usual definition of "Bayesian", as someone who does inference by combining likelihood and prior. Bayesians always have more than one hypothesis (outside trivial situations ...

21mo

A platonically perfect Bayesian given complete information and with accurate priors cannot be substantially fooled. But once again this is true regardless of whether I report p-values or likelihood ratios. p-values are fine.

*If I do an experiment, you generally don't know the precise alternate hypothesis in advance -- you want to test if the coin is fair, but you don't know precisely what bias it will have if it's unfair.*

Yes. But as far as I can see this isn't of any particular importance to this discussion. Why do you think it is?

*If we fix the two alternate hypotheses in advance, and if I have to report all data, then I'm reduced to only hacking by choosing the experiment that maximizes the chance of luckily passing your threshold via fluke. This is unlikely, as you say*...

31mo

It's the key of my point, but you're right that I should clarify the math here. Consider this part:
This is true for one hypothesis. It is NOT true if you know the alternative hypothesis. That is to say: suppose you are checking the p-value BOTH for the null hypothesis bias=0.5, AND for the alternate hypothesis bias=0.55. You check both p-values and see which is smaller. Now it is no longer true that you can keep collecting more data until their desired hypothesis wins; if the truth is bias=0.5, then after enough flips, the alternative hypothesis will never win again, and will always have astronomically small p-value.
To repeat: yes, you can disprove bias=0.5 with p<0.05; but at the time this happens, the alternative hypothesis of bias=0.55 might be disproven at p<10^{-100}. You are no longer guaranteed to win when there are two hypotheses rather than one.
Importantly, this is false! This statement is wrong if you have only one hypothesis rather than two.
More specifically, I claim that if a sequence of coin flip outcomes disproves bias=0.5 at some p-value p, then for the same sequence of coin flips, there exists a bias b such that the likelihood ratio between bias b and bias 0.5 is O(1/p):1. I'm not sure what the exact constant in the big-O notation is (I was trying to calculate it, and I think it's at most 10). Suppose it's 10. Then if you have p=0.001, you'll have likelihood ratio 100:1 for some bias.
Therefore, to get the likelihood ratio as high as you wish, you could employ the following strategy. First, flip coins until the p value is very low, as you described. Then stop, and analyze the sequence of coin flips to determine the special bias b in my claimed theorem above. Then publish a paper claiming "the bias of the coin is b rather than 0.5, here's my super high likelihood ratio". This is guaranteed to work (with enough coinflips).
(Generally, if the number of coin flips is N, the bias b will be on the order of 1/2±O(1/√N), so it will be pretty close

*I am saying that Yudkowsky is just plain wrong here, because omitting info is not the same as outright lying.*

This is silly. Obviously, Yudkowsky isn't going to go off on a tangent about all the ways people can lie indirectly, and how a Bayesian ought to account for such possibilities - that's not the topic. In a scientific paper, it is implicit that all relevant information must be disclosed - not doing so is lying. Similarly, a scientific journal must ethically publish papers based on quality, not conclusion. They're lying if they don't. As for auth...

11mo

You're welcome to play semantic games if you wish, but that's not how most people use the word "lying" and not how most people understand Yudkowsky's post.
By this token, p-values also can never be hacked, because doing so is lying. (I can just define lying to be anything that hacks the p-values, which is what you seem to be doing here when you say that not publishing a paper amounts to lying.)
You're switching goalposts. Yudkowsky was talking exclusively about how I can affect the likelihood ratio. You're switching to talking about how I can affect your posterior. Obviously, your posterior depends on your prior, so with sufficiently good prior you'll be right about everything. This is why I didn't understand you originally: you (a) used H for "hypothesis" instead of for "heads" as in the main post; and (b) used 0.9 for a posterior probability instead of using 10:1 for a likelihood ratio.
To the extent you're saying something true here, it is also true for p values. To the extent you're saying something that's not true for p values, it's also false for likelihood ratios (if I get to pick the alternate hypothesis).
No, they don't. That is precisely the point of p-hacking.
The stopping rule is not a central example of p-hacking and never was. But even for the stopping rule for coin flips, if you let me choose the alternate hypothesis instead of keeping it fixed, I can manipulate the likelihood ratio. And note that this is the more realistic scenario in real experiments! If I do an experiment, you generally don't know the precise alternate hypothesis in advance -- you want to test if the coin is fair, but you don't know precisely what bias it will have if it's unfair.
If we fix the two alternate hypotheses in advance, and if I have to report all data, then I'm reduced to only hacking by choosing the experiment that maximizes the chance of luckily passing your threshold via fluke. This is unlikely, as you say, so it's a weak form of "hacking". But this is also

If you say that you are reporting all your observations, but actually report only a favourable subset of them, and the Bayesian for some reason assigns low probability to you deceiving them in this way, when actually you are deceiving them, then the Bayesian will come to the wrong conclusion. I don't think this is surprising or controversial.

But I don't see how the Bayesian comes to a wrong conclusion if you truthfully report all your observations, even if they are taken according to some scheme that produces a distribution of likelihood ratios that is sup...

21mo

OK but please attempt to square this with Yudkowsky's claim:
I am saying that Yudkowsky is just plain wrong here, because omitting info is not the same as outright lying. And publication bias happens when the person omitting the info is not even the same one as the person publishing the study (null results are often never published).
This is just one way to p-hack a Bayesian; there are plenty of others, including the most common type of p-hack ever, the forking paths (e.g. this xkcd still works the same if you report likelihoods).
I'm not sure what you mean by "wrong conclusion" exactly, but I'll note that your statement here is more-or-less also true for p-values. The main difference is that p-values try to only convince you the null hypothesis is false, which is an easier task; the likelihood ratio tries to convince you some specific alternate hypothesis has higher likelihood, which is necessarily a harder task.
Even with Eliezer's original setup, in which the only thing I can control is when to stop the coin flip, it is hard to get p<0.001. Moreover, if I do manage to get p<0.001, that same sequence of coins will have a likelihood ratio of something like 100:1 in favor of the coin having a mild bias, if my calculation is correct. A large part of Eliezer's trick in his program's simulation is that he looked at the likelihood ratio of 50% heads vs 55% heads; such a specific choice of hypotheses is much harder to hack than if you let me choose the hypotheses after I saw the coinflips (I may need to compare the 50% to 60% or to 52% to get an impressive likelihood ratio, depending on the number of coins I flipped before stopping).
This is never the scenario, though. It is very easy to tell that the coin is not 90% biased no matter what statistics you use. The scenario is usually that my drug improves outcomes a little bit, and I'm not sure how much exactly. I want to convince you it improves outcomes, but we don't know in advance how much exactly they improve. P

I think that various "pro-fertility" people have a variety of motivations.

But "more people are better" ought to be a belief of everyone, whether pro-fertility or not. It's an "other things being equal" statement, of course - more people at no cost or other tradeoff is good. One can believe that and still think that less people would be a good idea in the current situation. But if you don't think more people are good when there's no tradeoff, I don't see what moral view you can have other than nihilism or some form of extreme egoism.

BTW: I...

Integrals of the likelihood function aren't really meaningful, even if normalized so the integral is one over the whole range. This is because the result depends on the arbitrary choice of parameterization - eg, whether you parameterize a probability by p in [0,1], or by log(p) in [-oo,0]. In Bayesian inference, one always integrates the likelihood only after multiplying by the prior, which can be seen as a specification of how the integration is to be done.

1mo40

I think you've got his pretty much figured out. But you may be missing an additional subtlety.

You say "Bayesian likelihood ratios really do only depend on the probability each hypothesis assigned only to the information that you received". Which could be interpreted as saying that the "likelihood function" is the probability assigned to the information received, seen as a function of *f*. But the likelihood function is actually not a function at all, but rather an equivalence class of functions of *f* that differ only by an overall pos...

11mo

Yeah, I discovered that part on accident at one point because I used the binomial distribution equation in a situation where it didn't really apply, but still got the right answer.
I would think the most natural way to write a likelihood function would be to divide by the integral from 0 to 1, so that the total area under the curve is 1. That way the integral from a to b gives the probability the hypothesis assigns to receiving a result between a and b. But all that really matters is the ratios, which stay the same even without that.

Your factual claims here seem at least somewhat reasonable. Naively extrapolating sub-replacement fertility centuries into the future is silly. Our wealthy civilization ought to be capable of finding some way of coping with increased elder care. The current number of humans may perhaps be more than is optimal.

But your moral view is atrocious. Human extinction would be bad - very bad. Because humans are the most interesting species to arise so far. Because human striving is properly focused on the future, not just hedonistic pl...

21mo

Worrying about extinction is one thing, and we're nowhere near that point, but does the pro fertility case rely on the philosophical assumption that more people is better? Surely you can see how some people might not find that very compelling.

I tested this with ChatGPT4 just now. It mostly got my specific questions right, but its commentary reveals some substantial misconceptions, such as that a flight from Ecuador to Sri Lanka would *first* pass over the Atlantic Ocean and *then* over Brazil.

Here it is:

User

If I flew by the most direct route from Ecuador to Sri Lanka, would I pass over Egypt?

ChatGPT

If you were to fly the most direct route (a great circle route) from Ecuador to Sri Lanka, you would not pass over Egypt.

Here's a general idea of what the path would look like:

- Starting from

Well, given that the text of the US constitution seems to clearly state that all powers not explicitly granted to the federal government belong to the states (or the people), I don't see how "power is devolved to the states from the federal government". It seems that the states don't need to wait for the federal government to "devolve" power to them in order to do something. As indeed we saw recently with respect to covid policy.

You could argue that the federal government "lets" the states do this, in the sense that the federal government has more guns tha...

33mo

Your textual interpretation of the constitution may be the right one, but it is not the one that governs as law in the US. Supreme Court has been very clear for the past 100+ years that the 10th amendment means literally nothing.

*the constitution is quite clear that power is devolved to the states from the federal government, and not that the federal government is granted power at the behest of the states*

The 10th amendment to the US constitution says:

The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

which sounds like the opposite of what you say. Of course, practice may be different.

23mo

My understanding is that since the Civil War the interpretation of the 10th Amendment is that states retain powers not because they are independent states participating with the federal government, but are explicitly subject to federal authority and maintain their powers only insofar as the constitution protects them and the federal government doesn't move to take them by rule of law. Prior to the end of the Civil War it seems to have generally been held that states were independently associated with the United States and could leave, and we fought a war to assert that they could not. This makes it clear that any powers they have are because the federal government lets them have them rather than the other way around.

*Can you tell me why?*

It think if we encountered aliens who were apparently not hostile, but presumably strange, and likely disgusting or disturbing in some ways, there would be three groups (likely overlapping) of people opposed to wiping them out:

- Those who see wiping them out as morally wrong.
- Those who see wiping them out as imprudent - we might fail, and then they wipe us out, or other aliens now see us as dangerous, and wipe us out.
- Those who see wiping them out as not profitable - better to trade with them.

There would also be three groups in favour of wi...

We are a species that has evolved in competition with other species. Yet, I think there is at least a 5% chance that if we encountered an intelligent alien species that we wouldn't try to wipe them out (unless they were trying to wipe us out).

Biological evolution of us and aliens would in itself be a commonality, that might produce some common values, whereas there need be no common values with an AI created by a much different process and not successfully aligned.

13mo

Biological evolution actively selects for values that we don't want whereas in AI training we actively select for values we do want. Alien life may not also use the biosphere the same way we do. The usual argument about common values is almost everything needs to breathe air, but at the same time competing and eliminating competing species is a common value among biological life.
Can you tell me why? We have wiped out every other intelligent species more or less. Subgroups of our species are also actively wiping out other subgroups of our species they don't like.

One problem I have with Diamond's theory is that I doubt that there is anything for it to explain. The Americas and Eurasia/Africa were essentially isolated from each other for about 15,000 years. In 1500 AD, the Americas were roughly 3500 years less advanced than Eurasia/Africa. That seems well within the random variation one would expect between two isolated instances of human cultural development over a 15,000 year time span. If you think there is still some remaining indication that the Americas were disadvantaged, the fact that the Americas are about half the size of Eurasia/Africa seems like a sufficient explanation.

*Perhaps you could give the definition you would use for the word "probability".*

I define it as one's personal degree of belief in a proposition, at the time the judgement of probability is being made. It has meaning only in so far it is (or may be) used to make a decision, or is part of a general world model that is itself meaningful. (For example, we might assign a probability to Jupiter having a solid core, even though that makes no difference to anything we plan to do, because that proposition is part of an overall theory of physics that is meaning...

*I think we actually have two quantities:*

*"Quobability" - The frequency of correct guesses made divided by the total number of guesses made.*

*"Srobability" - The frequency of trials in which the correct guess was made, divided by the number of trials.*

*Quabability is 1/3, Scrobability is 1/2. "Probability" is (I think) an under-precise term that could mean either of the two.*

I suspect that the real problem isn't with the word "probability", but rather the word "guess". In everyday usage, we use "guess" when the aim is to guess correctly. But the aim here is to no...

24mo

Maybe we are starting to go in circles. But while I agree the word "guess" might be problematic I think you still have an ambiguity with what the word probability means in this case. Perhaps you could give the definition you would use for the word "probability".
"In everyday usage, we use "guess" when the aim is to guess correctly." Guess correctly in the largest proportion of trials, or in the largest proportion of guesses? I think my "scrob" and "quob" thingies are indeed aiming to guess correctly. One in the most possible trials, the other in the most possible individual instances of making a guess.
"Having eliminated the word "guess", why would one think that Beauty's use of the strategy of randomly taking action H or action T with equal probabilities implies that she must have P(Heads)=1/2?" - I initially conjectured this as weak evidence, but no longer hold the position at all, as I explained in the post with the graph. However, I still think that in the other death-scenario (Guess Wrong you die) the fact that deterministically picking heads is equally good to deterministically picking tails says something. This GWYD case sets the rules of the wager such that Beauty is trying to be right in as many trials as possible, instead of for as many individual awakenings. Clearly moving the goalposts to the "number of trials" denominator.
For me, the issue is that you appear to take "probability" as "obviously" meaning "proportion of awakenings". I do not think this is forced on us by anything, and that both denominators (awakenings and trials) provide us with useful information that can beneficially inform our decision making, depending on whether we want to be right in as many awakenings or trials as possible. Perhaps you could explain your position while tabooing the word "probability"? Because, I think we have entered the Tree-falling-in-forest zone: https://www.lesswrong.com/posts/7X2j8HAkWdmMoS8PE/disputing-definitions, and I have tried to split our problem t

24mo

Yes I do, very good point!

*One could argue that if the coin is flicked and comes up tails then we have both "Tails&Monday" and "Tails&Tuesday" as both being correct, sequentially.*

Yes, it is a commonplace occurrence that "Today is Monday" and "Today is Tuesday" can both be true, on different days. This doesn't ordinarily prevent people from assigning probabilities to statements like "Today is Monday", when they happen to not remember for sure whether it is Monday or not **now**. And the situation is the same for Beauty - it is either Monday or Tuesday, she doesn't know which, but...

14mo

We are in complete agreement about how beauty should strategize given each of the three games (bet a dollar on the coin flick with odds K, GWYL and GWYD). The only difference is that you are insisting that "1/3" is Beauty's "degree of belief". (By the way I am glad you repeated the same maths I did for GWYD, it was simple enough but the answer felt surprising so I am glad you got the same.)
In contrast, I think we actually have two quantities:
"Quobability" - The frequency of correct guesses made divided by the total number of guesses made.
"Srobability" - The frequency of trials in which the correct guess was made, divided by the number of trials.
Quabability is 1/3, Scrobability is 1/2. "Probability" is (I think) an under-precise term that could mean either of the two.
You say you are a Bayesian, not a frequentist. So for you "probability" is degree of belief. I would also consider myself a Bayesian, and I would say that normally I can express my degree of belief with a single number, but that in this case I want to give two numbers, "Quobability = 1/3, Scrobability =1/2". What I like about giving two numbers is that typically a Bayesian's single probability value given is indicative of how they would bet. In this case the two quantities are both needed to see how I would bet given slightly varies betting rules.
I was still interested in "GRYL", which I had originally assumed would support the thirder position, but (for a normal coin) had the optimal tactic being to pick at 50/50 odds. I just looked at biased coins.
For a biased coin that (when flicked normally, without any sleep or anmesia) comes up heads with probability k. (k on x-axis), I assume Beauty is playing GRYL, and that she is guessing heads with some probability. The optimal probability for her strategy to take is on the y-axis (blue line). Overall chance of survival is orange line.
You are completely correct that her guess is in no way related to the actual coin bias (k), except for k=0.

*If the Tuesday bet is considered to be "you can take the bet, but it will replace the one you may or may not have given on a previous day if their was one", then things line up to half again.*

I can think of two interpretations of the setup you're describing here, but for both interpretations, Beauty does the right thing only if she thinks Heads has probability 1/3, **not** 1/2.

Note that depending on the context, a probability of 1/2 for something does not necessarily lead one to bet on it at 1:1 odds. For instance, if based on almost no knowledge of baseb...

24mo

"If Beauty instead thinks that P(Heads)=1/2, then P(Heads&Monday)=1/2. Trying to guess what a Halfer would think, I'll also assume that she thinks that P(Tails&Monday)=1/4 and P(Tails&Tuesday)=1/4"
This part of the analysis gets to the core of my opinion. One could argue that if the coin is flicked and comes up tails then we have both "Tails&Monday" and "Tails&Tuesday" as both being correct, sequentially. They are not mutually exclusive outcomes. One could also argue that, on a particular waking up Beauty doesn't know which day it is and thus they are exclusive in that moment.*
Which gets me back to my point. What do you mean by "probability". If we take a frequentist picture do we divide by the number of timelines (or equivalently separate experimental runs if many are run), or do we divide by the number of times beauty wakes up. I think either can be defended and that you should avoid the words "probability" or the notation P( X ) until after you have specified which denominator you are taking.
I like the idea of trying to evaluate the optimal policy for Beauty, given the rules of the game. Maybe you are asked to right a computer function (Beauty.py) that is going to play the game as described.
I liked the thought about indeteriminisitc strategies sometimes being the best. I cooked up an extreme example underneath here to (1) show how I think the "find the optimal policy approach" works in practice and to (2) give another example of randomness being a real friend.
----------------------------------------
Consider two very extreme cases of the sleeping beauty game:
Guess wrong and you die! (GWYD)
Guess right and you live! (GRYL)
In both instances beauty guesses whether the coin was heads of tails on each waking. After the experiment is over her guesses are considered. In the first game, one (or more) wrong guesses get beauty killed. In the Second a single right guess is needed, otherwise she is killed.
In the first game, GWYD, (I think) it is obvious

*I agree that you can make a betting/scoring setup such that betting/predicting at 50% is correct. Eg,suppose that on both Monday and Tuesday, Beauty gets to make a $1 bet at some odds. If she's asleep on Tuesday then whatever bet she made on Monday is repeated. In that case she should bet at 1:1 odds.*

Let's work it out if Beauty has 1/3 probability for Heads, with the 2/3 probability for Tails split evenly between 1/3 for Tails&Monday and 1/3 for Tails&Tuesday.

Here, the possible actions are "don't bet" or "bet on Heads", at 1:1 odds (let's say win $...

There are rules for how to do arithmetic. If you want to get the right answer, you have to follow them. So, when adding 18 and 17, you can't just decide that you don't like to carry 1s today, and hence compute that 18+17=25.

Similarly, there are rules for how to do Bayesian probability calculations. If you want to get the right answer, you have to follow them. One of the rules is that the posterior probability of something is found by conditioning on **all** the data you have. If you do a clinical trial with 1000 subjects, you can't just decide that you'd...

Yes, I can't get it either.

The possibility of a broken link (now or in the future) is one of several reasons why it would have been a good idea to mention the title of the paper and the authors' names.

*Nope. Doesn't work this way. There is an important difference between a probability of a specific low probable event happening and a probability of any low probable event from a huge class of events happening.*

In Bayesian probability theory, it certainly does work this way. To find the posterior probability of Heads, given what you have observed, you combine the prior probability with the likelihood for Heads vs. Tails based on **everything** that you have observed. You don't say, "but this observation is one of a large class of observations that I've decided t...

14mo

You are making a fascinating mistake, and I may make a separate post about it, even though it's not particularly related to anthropics and just a curious detail about probability theory, which in retrospect I relize I was confused myself about. I'd recommend you to meditate on it for a while. You already have all the information required to figure it out. You just need to switch yourself from the "argument mode" to "investigation mode".
Here are a couple more hints that you may find useful.
1) Suppose you observed number 71 on a random number generator that produces numbers from 0 to 99.
Is it
* 1 in 100 occurence because the number is exactly 71?
* 1 in 50 occurence becaue the number consist of these two digits: 7 and 1?
* 1 in 10 occurence because the first digit is 7?
* 1 in 2 occurence because the number is more or equal 50?
* 1 in n occurence because it's possible to come with some other arbitrary rule?
What determine which case is actually true?
2) Suppose you observed a list of numbers with length n, produced by this random number generator. The probability that exactly this series is produced is 1/100n
* At what n are you completely shocked and in total disbelief about your reality, after all you've just observed an event that your model of reality claims to be extremely improbable?
* Would you be more shocked if all the numbers in this list are the same? If so why?
* Can you now produce arbitrary improbable events just by having a random number generator? In what sense are these events have probability 1/100n if you can witness as many of them as you want any time?
You do not need to tell me the answers. It's just something I believe will be helpful for you to honestly think about.
Here is the last hint, actually I have a feeling that this just spoils the solution outright so it's in rot13:
Gur bofreingvbaf "Enaqbz ahzore trarengbe cebqhprq n ahzore" naq "Enaqbz ahzore trarengbe cebqhprq gur rknpg ahzore V'ir thrffrq" ner qvssrerag bofre

14mo

Tangent: I ran across an apparently Frequentist analysis of Sleeping Beauty here: Sleeping Beauty: Exploring a Neglected Solution, Luna
To make the concept meaningful under Frequentism, Luna has Beauty perform an experiment of asking the higher level experimenters which awakening she is in (H1, T1, or T2). If she undergoes both sets of experiments many times, the frequency of the experimenters responding H1 will tend to 1/3, and so the Frequentist probability is similarly 1/3.
I say "apparently Frequentist" because Luna doesn't use the term and I'm not sure of the exact terminology when Luna reasons about the frequency of hypothetical experiments that Beauty has not actually performed.

*Any other event, that the beauty is rightfully confident to be low probable but possible for both Monday and Tuesday, happens.*

And since this happens **absolutely every time she wakes**, Beauty should always assess the probability of Heads as 1/3.

There's always fly crawling on the wall in a random direction, unlikely to be the same on Monday and Tuesday, or a stray thought about aardvarks, or a dimming of the light from the window as a cloud passes overhead, or any of millions of other things entering her consciousness in ways that won't be the same Mond...

24mo

Nope. Doesn't work this way. There is an important difference between a probability of a specific low probable event happening and a probability of any low probable event from a huge class of events happening. Unsurprisingly, the former is much more probable than the latter and the trick works only with low probable events. As I've explicitly said, and you could've checked yourself as I provided you the code for it.
It's easy to see why something that happens absolutely every time she wakes doesn't help at all. You see, 50% of the coin tosses are Heads. If Beauty correctly guessed Tails 2/3 out of all experiments that would be a contradiction. But it's possible for the Beauty to correctly guess Tails 2/3 out of some subset of experiments. To get the 2/3 score she need some kind of evidence that happens more often when the coin is Tails than when it is Heads, not all the time, and then guess only when she gets this evidence.
This is irrelevant unless the Beauty somehow knows where the fly is supposed to be on Monday and where on Tuesday. She can try to guess tails when the fly is in a specific place that the beauty precommited to, hoping that the causal process that define fly position is close enough to placing the fly in this place with the same low probability for every day but it's not guaranteed to work.
I've linked two posts. You need to read the second one as well, to understand the mistake in the reasoning of the first.
This is exactly how probability theory works. The event "random number generator produced any number" has very high probability. The event "random number generator produced this specific number" has low probability. Which event we are talking about depends on whether the number was specified or not. It can be confusing if you forget that probabilities are in the mind, that it's about Beauty decision making process, not the metaphysical essence of randomness.
The fact that Beauty is unable to tell which day it is or whether she has been

I'm not trying to be disrespectful here, just trying to honestly state what I think (which I believe is more respectful than not doing so).

If I understand your position correctly, you think Beauty should rationally think as follows:

- Wakes up, notices the experimenters are not here to tell her it's Wednesday and the experiment is over. Concludes it's Monday or Tuesday.
- Thinks to herself, "so, I wonder if the coin landed Heads?".
- Thinks a bit, and decides the probability of Heads is 1/2.
- Looks up and sees a dead rat crash to the ground just outside the win

14mo

Here is how I think Beauty should rationally think:
* Beauty notices the experimenters are not here to tell her it's Wednesday and the experiment is over. Concludes it's Monday or Tuesday.
* Thinks to herself, "so, I wonder if the coin landed Heads?"
* (optional) Entertains the idea that her awakening itself is an evidence in favor of Tails, but realizes that it would be true only if her current experience was randomly sampled among three independent outcomes Tails&Monday, Tails&Tuesday and Heads&Monday, which isn't true for this experiment. Tails&Tuesday necessary follows Tails&Monday and the causal process that determines her state isn't random sample from these outcomes.
* Concludes that she doesn't have any evidence that would distinguish an outcome where coin landed Heads from the outcome where the coin landed Tails, thus keeps her prior 1/2.
* (either) Suddenly a prince bursts through the door and explains that he managed to overcome the defenses of the evil scientists keeping her here in an unlikely feat of martial prowess and is rescuing her now
* (or) The beauty has second thought about participating in the experiment. She has good reasons to stay but also some part of her wants to leave. Thankfully there is a random number generator in her room (she has another coin, for example), capable of outputting numbers in some huge interval. She thinks about a number and decides that if the generator output exactly it she will leave the room, otherwise she stays. The generator outputs the number the beauty has guessed. And she leaves.
* (or) Suddenly the beauty has panic attack that makes her leave the room immediately. She remembers that scientists told her that the experimental sleeping pills she took on Sunday have this rare side effects that can manifest in any person in 0.1% of cases in any day of the next week.
* (or) Any other event, that the beauty is rightfully confident to be low probable but possible for both Monday and Tuesday, happens. Actuall

*This opens a tangent discussion about determinism and whether the amnesia is supposed to return her to the exactly the same state as before, but thankfully we do not need to go there.*

To briefly go there... returning her to the exactly same state would violate the no-cloning theorem of quantum mechanics. See https://en.wikipedia.org/wiki/No-cloning_theorem

*I'm saying that some of her probabilities become meaningful, even though they were not before. Tails&Monday, Tails&Tuesday, Heads&Monday become three elementary outcomes for her when she *...

14mo

The state doesn't have to be the same up to the quantum mechanics just the same to the point that SB has the same thoughts for the same reasons and it's quite possible that QM isn't required for it. Nevertheless let's not pursue this line of inquiry anymore.
I claim that they indeed need to sum to 1 so that we could use the formal probability theory in this setting as Kholmogorov axioms require that probability of the whole sample space was equal to one. However, they do not, as they are not three independant elementary outcomes as long as the beauty participates in the experiment. Thus we can't define probability space thus can't lawfully use the mathematical apparatus of formal probability theory.
I don't think it's fair to say that I deny the basis of rational discussion or adopted a completely absurd position. Sometimes maps do not corrspond to the territory. This includes mathematical models. There are settings in which 2+2 is not 4. There are settings where operation of "addition" cannot be defined. Likewise there are settings where probabilities of some events can not be defined. I believe Mikaël Cozic noticed one of the firsts that SB setting is problematic for certain probabilities in his paper for Double Halfer's position and the idea wasn't dismissed as a priori absurd.
Actually, as you may see from the post I'm Halfer for incubator SB and Double Halfer for the classic one. And I quite understand what Thirders position in classic points to and do not have much problem with the Antropical Motte. On the contrary you claimed that all the arguments in favour of 1/2 are bad. I think, between two of us, you can be more justifiably claimed to be mentally "trapped in a position".
However, I don't think it's a fruitful direction of a discussion. I empathize how ridiculous and absurd I look from your perspective because that's exactly how you look from mine. I believe you can do the same. So let's focus on our empathy instead of outgroup mentality and keep the

*You don't contradict the claim "you can't lawfully use probability theory in this setting" by showing that you will be able to use probability theory if you break the setting up.*

The setting is only "broken up" when she decides to leave the room, but she can think about probabilities before that. Are you saying that once she decides to leave the room, her probabilities for *aspects of the external world* should change?

It is completely typical for people to form probability judgements regarding what would happen if they were hypothetically to do somethin...

14mo

Hmm. This opens a tangent discussion about determinism and whether the amnesia is supposed to return her to the exactly the same state as before, but thankfully we do not need to go there. We can just assume that a prince charming bursts into the room to rescue the beauty or something
I'm saying that some of her probabilities become meaningful, even though they were not before. Tails&Monday, Tails&Tuesday, Heads&Monday become three elementary outcomes for her when she is suddenly not participating in the experiment. But when she is going along the experiment, Tails&Tuesday always follows Tails&Monday - these outcomes are causally connected and if you treat them as if they are not you arrive to the wrong conclusion.
It's easy to show that SB that every day of the experiment has a small chance of being interrupted/change her mind and walk away can correctly guess Tails with about 2/3 accuracy when she was interrupted/changed her mind. But only with 1/2 accuracy otherwise.
def interruption(heads_chance=0.5, interrupt_chance=0.001):
days, coin = classic(heads_chance=heads_chance)
for day in days:
if interrupt_chance > random.random():
return day, coin
return None, coin
interrupted_coin_guess = []
not_interrupted_coin_guess = []
for n in range(100000):
day, coin = interruption()
if day is not None:
interrupted_coin_guess.append(coin == 'Heads')
else:
not_interrupted_coin_guess.append(coin == 'Heads')
print(interrupted_coin_guess.count(True)/len(interrupted_coin_guess))
# 0.3006993006993007
print(not_interrupted_coin_guess.count(True)/len(not_interrupted_coin_guess))
# 0.5017374846029823

(Heads, Tuesday) does not occur in conjunction with her waking up, assuming the experimenters behave in the way they say they will. So when she wakes up, it is not a possibility for what her current situation is. Which is what we're talking about.

4mo10

*you can't lawfully use probability theory in this setting assuming that Tails and Monday, Tails and Tuesday and Heads and Monday are three exhaustive and exclusive outcomes which probabilities are supposed to be added to 1*

Why not?

Suppose for simplicity that Beauty knows that the coin is flipped on Sunday, on a table in a room next to her bedroom, and that the flipped coin is left on the table, with either Heads or Tails showing.

Now, suppose Beauty wakes up (and knows it's not Wednesday, since otherwise she would have been told immediately tha...

14mo

You don't contradict the claim "you can't lawfully use probability theory in this setting" by showing that you will be able to use probability theory if you break the setting up.
Also if the Beauty changes her mind an decides to break the setting of the experiment on Tuesday wouldn't she already have done it on Monday?

24mo

This is still a misleading framing, confusing "what she will discover" with "what is (aka: what she is guaranteed to discover later)". (tails, monday), (tails, tuesday), and (heads, monday) are the only experiences she will have, but (heads, tuesday) is assumed to happen in her model of the universe, though she won't know it happened until she's woken on Wednesday.

4mo181

You haven't mentioned the strongest argument for not updating your beliefs: *You aren't dead yet*.

The beliefs you hold now have not had fatal consequences (so far, at least). The same is not guaranteed if you change them.

4mo162

Corollary: If you see death coming, or e.g. you have a near miss and know it was only by chance that you survived, then now’s a good time to change your beliefs. Which, actually, seems to be a thing people do. (Though there are other reasons for that.)

There are two ways one might try to figure out these probabilities. One is that, in whatever final situation is being considered, you figure out the probability from scratch, as if the question had never occurred to you before. The other is that as you experience things you update your probability for something, according to the the likelihood ratio obtained from what you just observed, and in that way obtain a probability in the final situation.

When figuring out the probability of Heads from scratch on Wednesday, I think everyone agrees that it should be ...

Let's look at the scenario where Beauty remembers her last awakening.

When Beauty is woken and not told it's Wednesday, she should think Heads has probability 1/3, Tails 2/3. She knows that if the coin landed Heads, it is Monday, and that she will next wake up on Wednesday without forgetting anything. She knows that if the coin landed Tails, then with probability 1/2 it is Monday and her memory will soon be erased, and with probability 1/2 it is Tuesday and she will next wake up on Wednesday without forgetting anything.

So waking up on Wed...

14mo

Indeed. This I understand.
And here I stopped following you. There is exactly one person in her epistemic situation (Waking up on Wednesday remembering the previous awakening) in both Heads and Tails worlds. According to both SSA and SIA no update has to happen.

*What's Beauty credence for Heads when she wakes on Wednesday and doesn't remember any of her awakenings on Monday/Tuesday? If it's 1/2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?*

Because of the memory erasure, I think it's best to regard Beauty on Monday, on Tuesday (if woken), and on Wednesday as different people, all "descended" from Beauty on Sunday. But if you want to think of them as the same person, the change from 1/3 to 1/2 is a consequence of the new in...

14mo

How does it work? Especially in the scenario where SB doesn't loose her memory of the last awakening. What kind of Bayesian update is it? Why are coins less likely to turn out to be Tails On Wednesday than on Monday or Tuesday? Does the universe retcon the result of the coin toss somehow?

I think it may not be correct to shuffle this off into a box labelled "adversarial example" as if it doesn't say anything central about the nature of current go AIs.

Go involves intuitive aspects (what moves "look right"), and tree search, and also something that might be seen as "theorem proving". An example theorem is "a group with two eyes is alive". Another is "a capture race between two groups, one with 23 liberties, the other with 22 liberties, will be won by the group with more liberties". Human players don't search the tree down to...

I think the cyclic group exploit could have been found by humans. The idea behind it (maybe it gets confused about liberties when a group is circular) would probably be in the top 1000 ideas for exploits that a group of humans would brainstorm. Then these would need to be tested. Finding a working implementation would be a matter of trial and error, maybe taking a week. So if you got 100 good human go players to test these 1000 ideas, the exploit would be found within ten weeks.

The main challenge might be to maintain morale, with th...

64mo

We'll never know now, of course, since now everyone knows about weird circular patterns as something to try, along with more famous older exploits of computer Go programs like ladders or capture races.
First, I'd note that in terms of AI safety, either on offense or defense, it is unhelpful even if there is some non-zero probability of a large coordinated human effort finding a specific exploit. If you do that on a 'safe' model playing defense, it is not enough to find a single exploit (and presumably then patch it), because it can be hacked by another exploit; this is the same reason why test-and-patch is insufficient for secure software. Great, you found the 'circle exploit' - but you didn't find the 'square exploit' or the 'triangle exploit', and so your model gets exploited anyway. And from the offense perspective of attacking a malign model to defeat it, you can't run this hypothetical at all because by the time you get a copy of it, it's too late.
So, it's mostly a moot point whether it could be done from the AI safety perspective. No matter how you spin it, hard-to-find-but-easy-to-exploit exploits in superhuman models is just bad news for AI safety.
OK, but could humans? I suspect that humans could find it (unaided by algorithms) with only relatively low probability, for a few reasons.
First, they didn't find it already; open-source Go programs like Leela Zero are something like 6 years old now (it took a year or two after AG to clone it), and have been enthusiastically used by Go players, many of which would be interested in 'anti-AI tactics' (just as computer chess had its 'anti-engine tactics' period) or could stumble across it just wanking around doing weird things like making big circles. (And 'Go players' here is a large number: Go is still one of the most popular board games in the world, and while DeepMind may have largely abandoned the field, it's not like East Asians in particular stopped being interested in it or researching it.) So we have a

Did you think it was interesting when AIs became better than all humans at go?

If so, shouldn't you be interested to learn that this is no longer true?

14mo

Well, I wasn't interested because AIs were better than humans at go, I was interested because it was evidence of a trend of AIs being better at humans at some tasks, for its future implications on AI capabilities.
So from this perspective, I guess this article would be a reminder that adversarial training is an unsolved problem for safety, as Gwern said above. Still doesn't feel like all there is to it though.

4mo20

My favourite (done just now, with the July 20 version of chatGPT4):

Model: GPT-4

Why is gold denser than uranium?

ChatGPT

There seems to be a confusion in your question. Actually, uranium is denser than gold.

As of my last training cut-off in September 2021, the density of uranium is approximately 19.1 g/cm³, while that of gold is around 19.3 g/cm³. These densities can vary slightly depending on the exact conditions (like temperature and pressure), but generally, gold is not denser than uranium.

Density of a substance is determined by its atomic structure ...

54mo

I wrote the following in custom instructions, under "How would you like ChatGPT to respond?":
After every answer, give a short reflection, like "I looked at this answer again and it looks good" or "I thought about this again and I think it could be better if ..." or "After looking this over again, this makes no sense because...".
I then had the following conversation:
Q: What is denser, gold or uranium?
A: Uranium is denser than gold. The density of gold is about 19.32 grams per cubic centimeter, while the density of uranium is about 19.1 grams per cubic centimeter.
Upon reflection, I see that I made an error in my initial response. The correct densities are indeed 19.32 g/cm³ for gold and 19.1 g/cm³ for uranium, but these values indicate that gold is denser than uranium, not the other way around. I apologize for the confusion, and I appreciate your patience.

24mo

I find it funny that GPT-4 finds the need to account for the possibility that the densities of uranium or gold might have changed as of September 2021.

Two points...

Like a number of people (including Elga), you're converting an almost-doable thought experiment into one that may be impossible in principle. If the experiment is done with some not-yet-invented but plausible memory-erasing drug, but is otherwise realistic, Beauty will *not* have the same experiences when woken Monday and when woken Tuesday. Various aspects of her sensed environment, as well as internal thoughts, will be different for the two awakenings. We just assume that none of these differences allow her to infer th...

I'm not sure what you mean by "Beauty's expected experience ... on Wednesday". There are two possibilities. On Sunday, she will expect that they have equal probabilities. But that says nothing more than that on Sunday she expects Heads and Tails to be equally likely. When woken on Monday or Tuesday, here expectation of what Wednesday will be like isn't necessarily the same.

I also can't figure out what you mean by saying that 1/3 is "absolutely correct", while also saying that there could be another answer.

14mo

What's Beauty credence for Heads when she wakes on Wednesday and doesn't remember any of her awakenings on Monday/Tuesday? If it's 1/2 what is the reason for the change from 1/3? The fact that she has forgotten the information about her awakening during the experiment?
Suppose SB wakes up on Wednesday. She is told that it's Wednesday and then left alone to dress before she will be asked for the last time to guess Heads or Tails. She doesn't remember any of her awakenings and her credence for Heads is once again 1/2. However she finds a piece of paper hidden in her pajamas from the past version of herself, experiencing an awakening. This past version managed to cheat and send a message to the future where she claims that her current credence for Heads is 1/3. Should she expect to successfully guess Tails with 2/3 probability?
Or suppose that SB is given memory loss drug only before she is awaken on Tuesday so that on Wednesday she always remembers her last awakenings though she doesn't know whether it was on Monday or Tuesday. Is her credence for Heads still 1/3?

Jul 16, 202311-1

I've reviewed again the argument in Section 2 of Elga's paper (https://www.princeton.edu/~adame/papers/sleeping/sleeping.pdf) and it seems valid to me. It doesn't have the same simplicity of some other arguments for 1/3, though, such as the betting arguments, and my argument in https://glizen.com/radfordneal/anth.abstract.html that the probability of anyone having your detailed experience on awakening is twice are large if you are woken twice.

There are many good arguments for 1/3. The arguments for 1/2 are all bad, and defending them forces peo...

44mo

I stand by my "valid answers for different questions" position, but it's pretty difficult to swallow "The arguments for 1/2 are all bad", without being clear that you're excluding the simplest of questions: if nothing changes from the setup, what is Beauty's expected experience of being told on Wednesday what the results were?
The chance of a fair coin coming up tails is 1/2. Beauty has no evidence on waking up to alter that, as she'd experience that either way.
There are OTHER questions, and betting-odds considerations that lead to 1/3 being absolutely correct. I don't dispute that in any way, but I do object to claims that it's the only answer, without specifying further which questions she's answering.

5mo10

Presumably the aim is to enhance human intellectual capabilities, not necessarily the level of innate human intelligence. Looking at it that way, improvements to education seem like a much more promising approach (which isn't to say that one shouldn't do both, of course).

One might object that people have been trying to improve education for millennia, so why would one think there's any low-hanging fruit to be had here? There are two reasons. One is that enhancing intellectual capabilities has not been the only goal of education (or even the pri...

Ahh, but without a purpose, how can one tell what the question actually is?

You could be asking about who you get by tracing back births and matings over the last 300,000 years that led to you. But do you then count people more than once if they show up more than once in this back trace?

Or are you really asking where your genetic material came from? It's quite possible that *none* of your genetic material came from one of your great-great-great-great-great-great-great-grandfathers. Genes aren't selected independently to be included or not in...

35mo

Not every question has to have a purpose! :) Imagine that this is intellectual interest only, at least to me.
No? This question would be trivially easy if you did lol.
No. I personally wanted a literal answer to my literal question.

Providing a sensible answer is dependent on arriving at a sensible interpretation of the question. I'll assume that it is aimed at understanding to what degree farming or non-farming lifestyles have had an influence on the selection of genes that you carry. I assume that "farming lifestyle" includes people who don't actually farm, but obtain food from farmers, one way or another.

On that basis, and assuming you are a typical inhabitant of a society that hasn't recently engaged in much hunting/gathering (maybe some fishing, but not dominant), I w...

15mo

This was not my question, but you're free to answer a different one! :)

25mo

This is an important insight, if that's the reason behind the question. If you break one's genetic heritage into 30 equal slices, 29 of which are hunter-gatherer, one of which is farming, and the last fragment is a rounding error too short for evolution to matter. You'll likely find that evolution is punctuated by reactions to large changes in what makes for fitness in the environment. The first slice (change from little cooperation and very low cultural adaptations to cooperative hunter-gatherer tribes with some amount of cultural knowledge transfer) likely saw a fair bit of change. The second through 29th slices saw continued adaptation to previous adaptations, but no major disruptions at the evolutionary/genetic level. The 30th slice saw a huge environmental change, and a somewhat different selection pressure.
The most recent 1/30 could EASILY have more impact than the 28 10000-year segments before it.

I don't think you can separate these phenomena like this. Thugs who aren't official police can intimidate political opponents of the government, and then not be prosecuted by the government. Thugs can cause chaos that somehow goes away when an organization or community stops opposing the government, or pays money to associates of the thugs, with the government again not prosecuting the thugs for extortion. In fact, I find it hard to imagine a democratic government becoming authoritarian without it employing some extra-governmental coercio...

26mo

The OP takes Hungary and Turkey as examples of countries that went through the problematic transition. Vox has a long article on Hungary and it doesn't speak about thugs being used in a significant way. As far as my memories goes that wasn't the case in Turkey either.
When governments coordinate with corporations they don't need to surveil everyone themselves. If you look at China, they give the corporations a lot of responsibility to monitor their users to keep their internet licenses.
France is censoring Rumble. The "tiktok"-ban bill would have been essentially a move to allow the US government to shut down many internet companies it doesn't like in a similar way that the Chinese can.
Extra-governmental violence is a factor in some states that transition to dictatorship but it's not an universal factor.
In the US the FBI seems to be pretty good at fighting violent extremism on both sides of the spectrum.

How many people have been killed in the US by right-wing protestors in the last four years? How many have been killed by left-wing protestors?

46mo

If you care about the numbers, why should I do the searching for them?
In any case, they are irrelevant to the argument about authoritarianism. Authoritarianism is about abuse of power coming from authority and protestors don't have power.
Death due to protestors and protests in general are bad, but they are a different kind of danger than abuse of government power.

I can't say with any certainty what exactly happened. Neither, it seems, can anyone else, to judge by https://www.forbes.com/sites/siladityaray/2022/02/23/canada-begins-to-release-frozen-bank-accounts-of-freedom-convoy-protestors/?sh=2745dbcc6364

Note that there is no reason to take statements by government officials at face value.

Even if you believe that the government intended only to freeze accounts of people actually at the protests, this is still a dangerous expansion of government power, considering that there were no court hearings, where peopl...

On the contrary, I think there is no norm against board members criticizing corporate direction.

I think it is accepted that a member of the board of a for-profit corporation might publicly say that they think the corporation's X division should be shut down, in order to concentrate investment in the Y division, since they think the future market for Y will be greater than for X, even though the rest of the board disagrees. This might be done to get shareholders on-side for this change of direction.

For a non-profit, criticism regarding whether the cor... (read more)