test
gwillen
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A 1-vector is just a regular vector. A 2-vector (or “bivector”) is a quantity associated with a two-dimensional “direction”, which is an oriented plane. And so on.
Ok, but how do you actually “and so on” the orientability here? I have not actually tried to picture how you orient a 3-vector in a higher space. And I’m suspicious about my analogy between 1-vector and 2-vector orientation until I can picture that. (You can orient a plane by picking one of the two halves it divides a 3-d volume into, but you normally orient a line by thinking about the ends, not the sides where it divides the plane. Does that matter?)
What’s up with all the foo-vectors?
This is an attempt to succinctly (hah) answer a question I keep having to refresh my memory about: What’s up with (vectors, bivectors, axial vectors / pseudovectors, multivectors, the cross product, etc.?) How do they relate to each other?
Multivectors
Multivectors or k-vectors are a generalization of vectors. Vectors have a length and a direction, and can be thought of as one-dimensional; k-vectors generalize vectors to arbitrary dimension k. In this framework, a scalar—a quantity without direction—can be thought of as a 0-vector. A 1-vector is just a regular vector. A 2-vector (or “bivector”) is a quantity associated with a two-dimensional “direction”, which is an oriented plane. And so on.
What does it mean for a plane to be “oriented”? It means we pick one side to be the “right side” and the other to be the “wrong side”. (In the same way, a vector is an “oriented line”, which has a “right end” where we draw the arrowhead.)
The exterior (“wedge”) product
We get multivectors from vectors using the exterior product, or “wedge product”. In the 3-dimensional setting, the wedge product smells almost exactly like the cross product—it takes two vectors and gives back a bivector, whose magnitude is the area of a parallelogram formed by those two vectors, and whose orientation depends on the relative directions of the two vectors. (I’m being deliberately vague here to avoid saying anything false; I could say “according to the right-hand rule” to get the general point across, but a later point will be that the left-right choice here is arbitrary, and could have been chosen the other way.)
Pseudovectors
In an n-dimensional space, a pseudovector (or axial vector) is an (n-1)-vector—that is, an n-minus-one-dimensional multivector. (A pseudoscalar is an n-vector.) Consider a 3-dimensional space: A bivector picks out two dimensions of it (an oriented plane), but picking two out of three dimensions leaves just one dimension remaining un-picked. So every bivector (a plane with magnitude and orientation) can be matched up with some vector (with the same magnitude, and pointing normal to the plane in the direction of its orientation.)
So in 3-dimensional space, a bivector is a pseudovector, because it is very nearly equivalent to a regular vector. (And a trivector is a pseudoscalar—there is only one possible basis-trivector, since there are only three dimensions and it has to span all of them. So a pseudoscalar only has a magnitude, and no meaningful direction, just like a regular scalar.)
Orientation
Why did I say “very nearly equivalent”—what’s the “pseudo” part about? This is trickier to explain, and while it will work fine as a refresher for myself, I don’t know if I will get it across fully to anybody else, but I’ll try.
Consider unit vectors pointing along the X, Y, and Z axes. Consider also a bivector X ^ Y, which has unit magnitude, and is oriented with the “right side” pointing in the same direction as our Z vector.
Now, flip the whole space around as though you are looking at it in a mirror. You can do this by e.g. negating any of our three vectors. In the resulting space, the X ^ Y bivector is now oriented in the opposite direction from the Z vector. (Thinking about the right-hand rule, consider that a right hand viewed in the mirror looks like a left hand. So if we apply the right-hand rule to the mirrored space, it will point in the opposite direction from how it pointed in the non-mirrored space.) If you take the “pseudovector” view of it—treating X ^ Y as something like a vector pointing along the Z axis, instead of a plane oriented towards the +Z axis—you will see where the “psuedo” comes from. Reflecting the space in a mirror causes the vector and the pseudovector, which pointed in the same direction before, to now point in opposite directions.
If you haven’t encountered this before, it’s probably going to seem like sophistry or handwaving, sorry. All I can say to that is, I promise this actually makes a difference, although I cannot adequately explain why at this time.
The cross product
This all comes around to why people say things like “the cross product doesn’t really give a vector!” Because if you look at the universe in a mirror, the result of the cross product does not behave like a vector. It will not appear mirrored like regular vectors, because its direction depends on handedness, and mirrors reverse handedness.
This also explains why sometimes people say “the cross product gives a bivector” and other people say “the cross product gives a pseudovector”. In 3-dimensional space, which is the only place the cross product is well-defined, the two are equivalent.
gwillen’s Shortform
Thanks for the update! This is really interesting to follow along with.
I would say it’s extremely unclear to me that the question “what is your probability that you are agent X” in an anthropic question like this is meaningful and has a well-defined answer? You said “there are practical reasons you’d like to know”, but you haven’t actually concretely specified what will be done with the information.
In the process of looking for something I had previously read about this, I found the following post:
https://www.lesswrong.com/posts/y7jZ9BLEeuNTzgAE5/the-anthropic-trilemma
Which seems to be asking a very similar question to the one you’re considering. (It mentions Ebborians, but postdates that post significantly.)
I then found the thing I was actually looking for: https://www.lesswrong.com/tag/sleeping-beauty-paradox
Which demonstrates why “what rent the belief is paying” is critical:
If Beauty’s bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half. If the bets get paid out once per awakening, acting as if the probability is one third has the best expected value.
Which says, to me, that the probability is not uniquely defined—in the sense that a probability is really a claim about what sort of bets you would take, but in this case the way the bet is structured around different individuals/worlds is what controls the apparent “probability” you should choose to bet with.
I’m surprised to hear you say that oratory is one of our weakest areas. Although I guess I actually think we’re pretty strong in a lot of areas, so maybe ‘weakest’ isn’t that bad. I have been quite moved by a lot of speeches at various Solstices, but I guess those were the best ones; certainly there’s variance.
I think we’re doing pretty well at the “first 80%” of the work, in a lot of ways, and are finding ourselves in the “last 80%”—the detail-oriented polishing that takes a lot of effort and has diminishing returns. (Which is not to say we shouldn’t do it! There are still returns to be had.)
I do not have a lot of evidence or detailed thinking to support this viewpoint, but I think I agree with you. I have the general sense that anthropic probabilities like this do not necessarily have well-defined values.
One thing I’m surprised I haven’t seen listed yet: Adjustment for boosters (and generally updating the vaccine adjustments, which I think are almost certainly too generous for 1- and 2-dose vaccination, once Omicron is circulating.)
Lately, I have mostly been using Microcovid as a guide for training my intuition about how important the various factors are. I don’t have a lot of confidence in the actual output of the model overall right now, since it doesn’t account for boosters, or for Omicron. I also have a general distrust of some of the model’s simplifying assumptions about how factors interact, although I don’t have anything better to substitute, other than my own intuitive judgement.
Is there some way to see these? Or are these the same sequences Ray says he’s planning to launch soon, and I should wait until then?
In other words, Delta lasts longer in the air than the original strain (which itself could last in the air for as long as three hours)
This is tangential, but it’s been fascinating to me to watch the spread of that “three hours” factoid. I believe it originally comes from this paper: https://www.medrxiv.org/content/10.1101/2020.03.09.20033217v1.full.pdf (Note that “HCoV-19” was referring to what is now designated “SARS-CoV-2″, i.e. COVID-19.)
I very gently criticized it at the time for having a quite misleading abstract, regarding the “three hours” figure:
https://twitter.com/gwillen/status/1237854407007457280 https://twitter.com/DrNeeltje/status/1237895661967642626 https://twitter.com/gwillen/status/1237896626355593221
This abstract, as far as I can tell, is where the “three hours” factoid originated, which has now become gospel that appears on public health pages like your link above. In reality, the abstract said they detected the virus in aerosols for “up to three hours” because that was the duration of the experiment. It’s obvious from graphs, as well as their computed half-life figure—which is coincidentally pretty close to three hours—that if they had continued the experiment, they would have kept detecting viable virus for a multiple of that time. (Of course, this figure is also kind of meaningless anyway, since the initial quantity of virus they’re experimenting on was arbitrary! Only the half-life is really meaningful, but that doesn’t make good headlines.)
It feels very weird to me to have “been there at the beginning” for this. I have no specific qualifications for interpreting this paper, beyond being smart and careful. It’s a little depressing (but that’s the COVID story.)
(I would love for someone to correct my story here, and find some other explanation for where this factoid came from, or argue that I’m wrong about it being misleading to the point of error. I really don’t like this story or what it implies.)
I can’t join LW folks, because I’m already the co-lead of a different team (this year we are called 404 Rage Not Found), but I want to strongly endorse the MIT Mystery Hunt as an activity. For people who aren’t familiar with it, you should check it out.
I feel like this is clearly frontpage material, so I would second Aella’s questions about what changes would make that make sense.
This seems surprisingly related to a thought I’ve had about the inefficiency of markets in consumer goods. When buying things, people usually only have room to optimize on a small number of taut constraints. One of them is going to be price; you get one, maybe two more if you’re lucky. If some product or service has more axes of variation than “what does it cost” and “does it work at all”, they will typically get almost negligible optimization pressure applied to them.
For example, the problem of buying an airplane ticket already has a bunch of taut constraints—the right place, the right time, the right price. The wifi on planes is generally garbage; whether or not there is wifi has risen up to the point where it gets a bit of optimization pressure, but how well the wifi works is way below the required bar. There is no incentive to improve it. Similarly the food on planes.
(To be honest, this applies to employment relationships, too, which might explain some things about how people generally find jobs to suck.)
I’m not Zvi obviously, but my model is something like this: Wordpress is very boring. It’s not “a thing”. Nobody blames Wordpress for content published on Wordpress; they blame whoever wrote it. Probably the fact that every Wordpress blog has its own style contributes to this; there is mostly no unified “Wordpress brand”, except the brand of people who haven’t gotten around to picking a real theme yet.
Substack talks a good game about anti-censorship, but Substack is NOT boring. They are trying to make a name for themselves, they force common branding across all the blogs on their site, and they generally want Substack itself to be “a thing”, not just a neutral platform that fades into the background. And they are likely to attract controversy, and then as a result attract pressure. Usually people eventually fold under pressure. Promises not to fold under pressure are just meaningless marketing copy.
Would love an update if you do!
Thanks for the info!
I’m having trouble discerning this from your description and I’m curious—is this approach closely related to the approach GWS describes above, involving the beta distribution, which basically seems to amount to adding one “phantom success” and one “phantom failure” to the total tally?
Thank you, that is super interesting and informative.
I am wondering if what we’re seeing here is cross-reactivity. (I’m not sure if that’s the right term for it, but: repeatable false positives from the test reacting to some antigen that is “close enough” to the covid antigen it’s looking for.) I recall seeing a table of things that one of the tests was checked for cross-reactivity against—they generally aren’t good at distinguishing SARS from covid, they mostly don’t react to other stuff, but the rate is not zero. (It’s possible the thing I am thinking of was an isothermal NAAT test, not antigen.)
Looking for info about this kind of thing with antigen tests, I eventually found this:
https://onlinelibrary.wiley.com/doi/10.1111/ped.14582
They have three case studies of antigen test false-positives, in Japan, in children infected with Human Rhinovirus A, and one of them remained positive on a test from a different production batch (although not a different brand/test.) And they mention that other countries have reported cross-reactivity with “other coronaviruses, influenza virus, and Mycoplasma pneumoniae”.
If it eventually went away and didn’t come back, I think after reading this I’m going to put my money on “cross-reactivity to another asymptomatic viral infection.” If it didn’t go away, and you still test positive on further tests, that becomes even more interesting and maybe you could get someone to write a case study about it in exchange for figuring out what the heck it is.
Upvoted for thoughtful dissent and outside perspective.
I … have some complicated mixed feelings here. LW has a very substantial contingent of “gifted kids”, who spent a decent chunk of their (...I suppose I should say “our”) lives being frustrated that the world would not take them seriously due to age. Groups like that are never going to tolerate norms saying that young age is a reason to talk down to someone. And guidelines for protecting younger people from older people, to the extent that they involve disapproval or prevention of apparently-consensual choices by younger people, are going to be tricky that way. Any concern that “young minds are not allowed to waive” will be (rightly) seen as condescending, especially if you extend “young” to age 30. This does not really become less true if the concern is accurate.
This is extra-true here, because the “rationalist community” is not a single organization with a hierarchy, or indeed (I claim) even really a single community. So you can’t make enforceable global rules of conduct, and it’s very hard to kick someone out entirely (although I would say it’s effectively been done a couple of times.)
You might be relieved to learn that, at least from where I’m standing, a substantial fraction of the community is not in fact working towards (or necessarily even believing strongly in) the higher goal of preventing the AI apocalypse. (I am not personally working towards it; I would not say that I have a firm resolution either way on how much I believe in it, but I tend towards being skeptical of most specific forms that I have seen described.)
And, not to “tu quoque” exactly, I hope, but… my sense is that academia is not great along this axis? I have never been a grad student, but I would say at least half my grad student friends have had significant mental health problems directly related to their work. And a small but substantial number have had larger problems stemming directly from abusive or (more often) incompetent advisors. In most cases, the latter seemed to have very little recourse against their advisors, especially the truly abusive ones, which seems like exactly the sort of thing that you’re calling out here. There were always theoretically paths they could take to deal with the problem, but in practice the advisor has so much more power in the relationship that it would usually involve major bridge-burning to use them, and in some cases it’s not clear it would have helped even then.
This latter problem—of theoretical escalation paths around your manager existing, but being unusable in practice—seems pretty similar, to me, between academia and industry. But my impression is that academia has much worse “managers”, on average, because advisors are selected primarily for research skill, and often have poor management skills.
This is all to say—coming back around to the point—that I think academia has lots of people who behave in ways similar to how Michael Vassar is described here. (I have not met him personally, and cannot speak to that description myself.) Granted, academia has rules of conduct that would prevent some of the things seen here. I expect it would be very rare for an advisor to get their advisees into psychedelic drugs. But on the flip side, people in Vassar’s “orbit” who grow disillusioned with him are free to leave. Grad students generally cannot do that, without a significant risk of losing years of work, and their hopes of an academic career.
If anything, I think the ability to say “this person is a terrible influence, and also we can acknowledge the good they have done” may be protective from a failure mode that I have anecdotally heard of in academia multiple times: the PI who is abusive in some way, and the “grapevine” is somewhat aware of this, but whose work is too valuable (e.g. in terms of grant money) to do anything about.
It seems to be “responsive”; I wasn’t seeing it on desktop either, until I made the window wider.