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
Karma: 317
This two-point distribution is important as the distribution where Markov’s inequality is an equality, so it’s cool to have it visualized as part of the proof.
is the correction to the probability density function really what you want, and are other deviations from Gaussianity expressible with cumulants? All I can think of is that the Gaussian is the maximum entropy distribution so maybe there’s a formula for how far below the maximum entropy you are. I don’t know what it’d be good for though.
I’m in the process of turning the ideas in a stack of my notebooks into what I hope will be a short paper, which is just one illustration of what I think was the real trade-off, which is between conciseness and time spent writing. Or for another, see the polished 20-page papers on logical decision theory. Though it’s not the same, they cover much of the same ground as the older expositions of timeless decision theory and updateless decision theory. There was a long period where these kinds of decision theories were only available through posts, and then through Eliezer Yudkowsky’s long TDT paper. That period could not have been skipped, and could only have been shortened in the sense that the same work could have been done faster at the expense of other work. See also this exchange on Twitter, though they’re not talking about being concise specifically:
Miles Brundage: I can’t speak to the details of those experiments but I at least read a much higher fraction of your paper outputs than your blog post outputs. Possibly I’m a minority here but I am certainly not the only one.
Eliezer Yudkowsky: Yeah, I tried it and it was way too fucking time-expensive. My guess is that 100x the output you like less ends up having the larger impact on the world.
Yes, and I’m realizing I went into a digression that wasn’t really relevant to my original point. In this particular post I just wanted to discuss the first principles calculation, that tells you that the sunlight hitting a relatively small area can supply all our electricity needs. The fact that just the area on roofs even makes a dent is one of the things that makes sense from this perspective, since roof area is not that large. Where to put solar panels is an economic question that doesn’t particularly matter for any of the points I’m going to make in this sequence, although I do want to get into the economics of batteries in some detail in the next two posts because that’s one of the things that limits how much solar capacity you can install. And, yes, the other big limitations are transmission and permitting—that’s a relevant point and I see now that you were trying to communicate how these other limitations can be addressed. I won’t really be getting to transmission and permitting, because this sequence was prompted by considering how I should update on battery storage exceeding expectations.
I’m replying to a post that said they lacked energy despite iron supplementation. Maybe it wasn’t iron deficiency, or maybe it could have been solved by raising the supplement dose, I don’t know, but if it was iron deficiency and the supplements weren’t helping then it’s good to know you can get iron in the same form as in meat from Impossible burgers.
Yeah, this is a US-centric perspective of mine but there’s no shortage of land here. This sounds to me like classic thriftiness about that which is not scarce, which isn’t real thriftiness. I mean, “effective use of both farmland and rooftops”… rooftops? What’s scarce here in the US is labor, not land. Why have all these people climbing up on rooftops? An interesting contrast is Texas (mostly utility solar) vs California (lots of rooftop solar). The interesting number, which I don’t know off the top of my head, is how many people are employed in the solar sector per unit capacity installed. I seem to remember California employs a lot more people, disproportionately to how much more solar it has.
The heme iron in meat is absorbed better than the non-heme iron in iron supplements, but Impossible Burger has heme iron. It’s very frustrating that this biotech advance of in vitro heme production is so far only used for this specific brand of meat substitutes but that’s the situation. I’m not sure why iron supplements didn’t work for you, as that same paper shows that even non-heme iron is absorbed well when blood iron is low, but maybe it depends on the individual? In any case, and I promise the company isn’t paying me to say this, I recommend Impossible burgers to vegans. It has to be this specific brand, they have the patent. There was another company with a heme production patent but they recently shut down.
The natural gas generation capacity that you need to cover for solar when it’s cloudy is, of course, less than what is required to make up for loss of solar after sundown.
Sure. There’s enough sunlight to run the whole country, so it’s physically possible, but it’s not at the moment technologically or economically practical, and may not be our best option in the near future. Until this wave of battery installations, though, I thought even California had saturated its solar potential. In the next post I’ll write in more detail about what I think is now possible, but briefly, it’s now feasible for all western US peak load (the extra power needed while people are awake) to be provided by solar and batteries. Whether we’ll also use solar for base load, and whether we’ll use it in cloudy areas, is a more difficult question that requires extrapolating prices, and I’ll try to address that in the third post.
I finally googled what Elon Musk has said about solar power, and found that he did a similar calculation recently on twitter:
Once you understand Kardashev Scale, it becomes utterly obvious that essentially all energy generation will be solar.
Also, just do the math on solar on Earth and you soon figure out that a relatively small corner of Texas or New Mexico can easily serve all US electricity.
One square mile on the surface receives ~2.5 Gigawatts of solar energy. That’s Gigawatts with a “G”. It’s ~30% higher in space. The Starlink global satellite network is entirely solar/battery powered.
Factoring in solar panel efficiency (25%), packing density (80%) and usable daylight hours (~6), a reasonable rule of thumb is 3GWh of energy per square mile per day. Easy math, but almost no one does these basic calculations.
Sort of. I think the distribution of Θ is the Ap distribution, since it satisfies that formula; Θ=p is Ap. It’s just that Jaynes prefers an exposition modeled on propositional logic, whereas a standard probability textbook begins with the definition of “random variables” like Θ, but this seems to me just a notational difference, since an equation like Θ=p is after all a proposition from the perspective of propositional logic. So I would rather say that Bayesian statisticians are in fact using it, and I was just explaining why you don’t find any exposition of it under that name. I don’t think there’s a real conceptual difference. Jaynes of course would object to the word “random” in “random variable” but it’s just a word, in my post I call it an “unknown quantity” and mathematically define it the usual way.
In solar-heavy areas before batteries (and without hydro), electricity in the early evening was provided by natural gas peaker plants, which can and do quickly shut off. Consider a scenario with growing demand. Prices in the early evening have to get pretty high before it’s worth paying for a whole natural gas plant just to run it for only a few hours.
I seen the argument made by Robert Bryce and Alex Epstein, who don’t suggest economic models, but the reason it’s at least not obvious to me is that we need to consider supply, demand, price as functions of time. Solar produces a glut of electricity during the day. It makes sense to me that it would increase the price of electricity in the early evening, when solar isn’t generating but demand is still high. It would do so by reducing the profitability of building natural gas plants to supply those hours, which results in either fewer natural gas plants (if demand is elastic) or the prices rising until they’re profitable (if demand is inelastic). How this affects the average price throughout the whole day I don’t know.
Yes, Tesla’s role in battery storage is actually odd and niche—their importance seems to be reducing the skill level required to built a battery energy storage facility by packaging batteries into self contained modules that contain all of the required equipment (thermal management, inverter, etc). The battery cells come from Asia.
The question to which Musk is relevant is not “how did we get to a world where battery storage is feasible” but “why would someone be investing in this ten or fifteen years ago when the technology was not there”. It seems to me to be a case where a futurist calculation that ignores the engineering details plus seemingly naive reasoning that the rest is “just engineering” would have actually given the right answer.
Elon Musk and Solar Futurism
This is a good example of neglecting magnitudes of effects. I think in this case most people just don’t know the magnitude, and wouldn’t really defend their answer in this way. It’s worth considering why people sometimes do continue to emphasize that an effect is not literally zero, even when it is effectively zero on the relevant scale.
I think it’s particularly common with risks. And the reason is that when someone doesn’t want you to do something, but doesn’t think their real reason will convince you, they often tell you it’s risky. Sometimes this gives them a motive to repeat superstitions. But sometimes, they report real but small risks.
For example, consider Matthew Yglesias on the harms of marijuana:
Inhaling smoke into your lungs is, pretty obviously, not a healthy activity. But beyond that, when Ally Memedovich, Laura E. Dowsett, Eldon Spackman, Tom Noseworthy, and Fiona Clement put together a meta-analysis to advise the Canadian government, they found evidence across studies of “increased risk of stroke and testicular cancer, brain changes that could affect learning and memory, and a particularly consistent link between cannabis use and mental illnesses involving psychosis.”
I’ll ignore the associations with mental illness, which are known to be the result of confounding, although this is itself an interesting category of fake risks. For example a mother that doesn’t want her child to get a tattoo, because poor people get tattoos, could likely find a correlation with poverty, or with any of the bad outcomes associated with poverty.
Let’s focus on testicular cancer, and assume for the moment that this one is not some kind of confounding, but is actually caused by smoking marijuana. The magnitude of the association:
The strongest association was found for non-seminoma development – for example, those using cannabis on at least a weekly basis had two and a half times greater odds of developing a non-seminoma TGCT compared those who never used cannabis (OR: 2.59, 95 % CI 1.60–4.19). We found inconclusive evidence regarding the relationship between cannabis use and the development of seminoma tumours.
What we really want is a relative risk (how much more likely is testicular cancer among smokers?) but for a rare outcome like testicular cancer, the odds ratio should approximate that. And testicular cancer is rare:
Testicular cancer is not common: about 1 of every 250 males will develop testicular cancer at some point during their lifetime.
So while doubling your testicular cancer risk sounds bad, doubling a small risk results in a small risk. I have called this a “homeopathic” increase, which is perhaps unfair; I should probably reserve that for probabilities on the order of homeopathic concentrations.
But it does seem to me to be psychologically like homeopathy. All that matters is to establish that a risk is present, it doesn’t particularly matter its size.
Although this risk is not nothing… it’s small but perhaps not negligible.
It’s great to have a LessWrong post that states the relationship between expected quality and a noisy measurement of quality:
(Why 0.5? Remember that performance is a sum of two random variables with standard deviation 1: the quality of the intervention and the noise of the trial. So when you see a performance number like 4, in expectation the quality of the intervention is 2 and the contribution from the noise of the trial (i.e. how lucky you got in the RCT) is also 2.)
We previously had a popular post on this topic, the tails come apart post, but it actually made a subtle mistake when stating this relationship. It says:
For concreteness (and granting normality), an R-square of 0.5 (corresponding to an angle of sixty degrees) means that +4SD (~1/15000) on a factor will be expected to be ‘merely’ +2SD (~1/40) in the outcome—and an R-square of 0.5 is remarkably strong in the social sciences, implying it accounts for half the variance.
The example under discussion in this quote is the same as the example in this post, where quality and noise have the same variance, and thus R^2=0.5. And superficially it seems to be stating the same thing: the expectation of quality is half the measurement.
But actually, this newer post is correct, and the older post is wrong. The key is that “Quality” and “Performance” in this post are not measured in standard deviations. Their standard deviations are 1 and √2, respectively. Elaborating on that: Quality has a variance, and standard deviation, of 1. The variance of Performance is the sum of the variances of Quality and noise, which is 2, and thus its standard deviation is √2. Now that we know their standard deviations, we can scale them to units of standard deviation, and obtain Quality (unchanged) and Performance/√2. The relationship between them is:
That is equivalent to the relationship stated in this post.
More generally, notating the variables in units of standard deviation as and (since they are “z-scores”),
where is the correlation coefficient. So if your noisy measurement of quality is standard deviations above its mean, then the expectation of quality is standard deviations above its mean. It is that is variance explained, and is thus 1⁄2 when the signal and noise have the same variance. That’s why in the example in this post, we divide the raw performance by 2, rather than converting it to standard deviations and dividing by 2.
I think it’s important to understand the relationship between the expected value of an unknown and the value of a noisy measurement of it, so it’s nice to see a whole post about this relationship. I do think it’s worth explicitly stating the relationship on a standard deviation scale, which this post doesn’t do, but I’ve done that here in my comment.
Some other comments brought up that the heme iron in meat is better absorbed, which is true, see figure 1. But the good news is that Impossible burgers have heme iron. They make it in yeast by adding a plasmid with the heme biosynthesis enzymes, pathway in Figure 1 of their patent on the 56th page of the pdf.
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.