2026-03-29 00:03:04
I’m excited to present my ambitious (since I built it myself) project—a website featuring FAQs on the most important topics for humanity (in my humble opinion).
The first FAQ on Immortalism is now available, just as I promised in previous posts.
https://humanitytomorrow.site/
It’s worth noting that the site is in demo mode—there may still be many bugs, the mobile and desktop versions differ (desktop is better), and I will be making many changes and improvements.
The most important part is the text, which is almost completely ready. Links to sources for strong or precise claims will be added in the coming days.
Currently, the FAQ focuses entirely on myths, misconceptions, objections, etc. In the future, a section covering scientific fields will be added.
The next section, after I finish working on the current one, will be an FAQ on AI risks.
The site is intended for those who are unfamiliar with these topics.
2026-03-28 23:00:13
Superintelligence, pp. 122–3. 2014.
Consider a technologically mature civilization capable of building sophisticated von Neumann probes of the kind discussed in the text. If these can travel at 50% of the speed of light, they can reach some
If we assume that 10% of stars have a planet that is—or could by means of terraforming be rendered—suitable for habitation by human-like creatures, and that it could then be home to a population of a billion individuals for a billion years (with a human life lasting a century), this suggests that around
There are, however, reasons to think this greatly underestimates the true number. By disassembling non-habitable planets and collecting matter from the interstellar medium, and using this material to construct Earth-like planets, or by increasing population densities, the number could be increased by at least a couple of orders of magnitude. And if instead of using the surfaces of solid planets, the future civilization built O'Neill cylinders, then many further orders of magnitude could be added, yielding a total of perhaps
Many more orders of magnitude of human-like beings could exist if we countenance digital implementations of minds—as we should. To calculate how many such digital minds could be created, we must estimate the computational power attainable by a technologically mature civilization. This is hard to do with any precision, but we can get a lower bound from technological designs that have been outlined in the literature. One such design builds on the idea of a Dyson sphere, a hypothetical system (described by the physicist Freeman Dyson in 1960) that would capture most of the energy output of a star by surrounding it with a system of solar-collecting structures. For a star like our Sun, this would generate
Combining these estimates with our earlier estimate of the number of stars that could be colonized, we get a number of about
It might not be immediately obvious to some readers why the ability to perform
In other words, assuming that the observable universe is void of extraterrestrial civilizations, then what hangs in the balance is at least 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 human lives (though the true number is probably larger). If we represent all the happiness experienced during one entire such life with a single teardrop of joy, then the happiness of these souls could fill and refill the Earth’s oceans every second, and keep doing so for a hundred billion billion millennia. It is really important that we make sure these truly are tears of joy.
2026-03-28 22:22:04
A landmark of social psychology research was “The Milgram Experiment,” but a new look at the audio tapes and other evidence collected during that experiment suggests that we may have been interpreting it incorrectly. Here is the Wikipedia summary of the experiment, showing how it is typically portrayed:
Yale University psychologist Stanley Milgram… intended to measure the willingness of study participants to obey an authority figure who instructed them to perform acts conflicting with their personal conscience. Participants were led to believe that they were assisting in a fictitious experiment, in which they had to administer electric shocks to a "learner". These fake electric shocks gradually increased to levels that would have been fatal had they been real.
The experiments unexpectedly found that a very high proportion of subjects would fully obey the instructions, with every participant going up to 300 volts, and 65% going up to the full 450 volts.
I don’t know about that “unexpectedly” part. I think the researchers suspected, in the wake of e.g. the Holocaust, that people were generally willing to obey awful instructions in ways that they failed to account for. Their experiment was designed to answer not whether but how much.
The results have since been interpreted as a kind of cynicism or caution about human nature, and about people’s tendencies to let their consciences be silenced by the trappings of authority.
But such takes may have been too optimistic.
Milgram interviewed his subjects after the experiment and found that those who stopped giving shocks felt that they were responsible for what they were doing, while those who continued giving shocks felt that the experimenter (the one giving the instructions to the subject) was responsible. Milgram theorized that his subjects, in the presence of an authority figure, stepped into a corresponding role: the “agentic state.” Once you are in that state, you stop considering yourself responsible for what you are doing and for the effects of what you are doing, and judge your actions only on whether you are doing it according to how the authority wants it done.
Arne Johan Vetlesen, in Evil and Human Agency (2005), pointed out that there is another possible interpretation: Milgram’s subjects may have had genuine sadistic impulses. In subjecting their victims to pain, they were not being somehow coerced by their situation to do things they would ordinarily not want to do, but that they were being allowed by their situation to do things they were ordinarily inhibited from doing.
He quoted Ernest Becker, who took a second look at Freud’s take on mob violence:[1]
…[M]an brings his motives in with him when he identifies with power figures. He is suggestible and submissive because he is waiting for the magical helper. He gives in to the magic transformation of the group because he wants relief of conflict and guilt. He follows the leader’s initiatory act because he needs priority magic so that he can delight in holy aggression. He moves in to kill the sacrificial scapegoat with the wave of the crowd, not because he is carried along by the wave, but because he likes the psychological barter of another life for his own: “You die, not me.” The motives and the needs are in men and not in situations or surroundings.
And Hannah Arendt, whose examination of the Adolf Eichmann trial was going on at around the same time as the early Milgram experiments, warned that the excuse of “obedience” (as used by the compliant Milgram subjects to explain their actions after-the-fact, and secondarily by Milgram himself in his theory) was not an explanation but a “fallacy”:
Only a child obeys. An adult actually supports the laws or the authority that claims obedience.[2]
Now David Kaposi and David Sumeghy have gone back through the audio tapes and other documentation preserved from the original Milgram experiments.[3]
What they found was that the “obedient” subjects were not in fact very compliant at all. Indeed none of them actually followed the experimental procedures they had been instructed to comply with.
Only a few of the subjects complied with the experimental procedures they were given in full, and all of them were among those who eventually refused to continue with the experiment.
(1) no entirely “fully obedient” participant fully obeyed the procedures Milgram’s experimenter instructed them to do; (2) violations of the procedures occurred on average 48.4% of the time in “fully obedient” sessions; and (3) violations occurred significantly more frequently in “fully obedient” than in the obedient phase of “disobedient” sessions.
Tellingly, these procedural violations were not efforts to avoid giving shocks, but actually increased the likelihood that an opportunity to give another shock would arise:
The most frequent violation in obedient sessions involved reading the memory test questions over the simulated screams of the learner. Doing this effectively guaranteed that the learner would fail the test and receive another shock. By talking over the protests, the obedient subjects abandoned the [ostensible] goal of testing memory and simply facilitated continuous shocks.[4]
The implication is that when Milgram interviewed the “obedient” subjects after the experiment was over, these subjects represented themselves as having merely obeyed because this was an excuse for their behavior that had been dangled before them temptingly during the experiment, and they anticipated that this excuse would be accepted. Milgram, by being willing to accept this excuse at face-value, in effect validated it and cooperated with the subjects in whitewashing their surrender to sadistic temptation.
See also H.L. Mencken (Damn: A book of Calumny) making a similar point about the supposedly hypnotic influence of the mob:
The numskull runs amuck in a crowd, not because he has been inoculated with new rascality by the mysterious crowd influence, but because his habitual rascality now has its only chance to function safely. In other words, the numskull is vicious, but a poltroon. He refrains from all attempts at lynching a cappella, not because it takes suggestion to make him desire to lynch, but because it takes the protection of a crowd to make him brave enough to try it.
⋮
In other words, the particular swinishness of a crowd is permanently resident in the majority of its members — in all those members, that is, who are naturally ignorant and vicious — perhaps 95 per cent. All studies of mob psychology are defective in that they underestimate this viciousness. They are poisoned by the prevailing delusion that the lower orders of men are angels. This is nonsense. The lower orders of men are incurable rascals, either individually or collectively. Decency, self-restraint, the sense of justice, courage — these virtues belong only to a small minority of men. This minority never runs amuck. Its most distinguishing character, in truth, is its resistance to all running amuck. The third-rate man, though he may wear the false whiskers of a first-rate man, may always be detected by his inability to keep his head in the face of an appeal to his emotions. A whoop strips off his disguise.
Moral Responsibility under Totalitarian Dictatorships
David Kaposi & David Sumeghy “From legitimate to illegitimate violence: Violations of the experimenter's instructions in Stanley Milgram’s ‘obedience to authority’ studies” Political Psychology (2026)
Karina Petrova “Audio tapes reveal mass rule-breaking in Milgram’s obedience experiments” PsyPost 28 March 2026
2026-03-28 19:14:18
As you are supposed to ask your doctor about supplements, I just asked my primary care doctor about taking glycine and N-acetylcysteine.
I had a lot more infections in the last two years and I'm doing some fascia work that could increase collagen turnover, so there's reasoning in that time frame that would be a causal explanation for increased glycine usage for collagen generation and less leftover for glutathione production.
My doctor just said, she doesn't know of any evidence for either supplement helping with immune function.
Should I fault her for not knowing about De Flora et al 1997 finding that N-acetylcysteine helps prophylactically to prevent influenza from becoming symptomatic? Should I fault her for not knowing that chicken bone broth soup has been shown in the last evidence review to be helpful for reducing the length of acute respiratory tract infections, and its glycine content being a prime causal explanation?
Should I fault her for not knowing that much about glutathione, as it doesn't come up that much in her daily practice?
Are you supposed to have the papers printed out to hand them to your doctor, or what are you supposed to do in a situation like that?
If you are doing are thinking about taking any non-standard interventions that are far away from the normal practice of medicine, why would you assume that the average doctor can tell you about the merit of the intervention?
Chatbots aren't perfect but they have access to a lot more information and if you talk to them about specific studies they can easily engage with you. You can push back. You can continue a conversation over multiple days when you first conversation made you more aware of other symptoms. You can ask another chatbot to check the work of the first one.
2026-03-28 15:07:01
(crosspost from my substack)
There’s almost nothing I love more than seeing a new scientific tool solve an old problem of life. Curing diseases, solutions to obesity, artificial wombs, brain uploading, cryonics, the journey from scarcity to abundance to immortality. I am infinitely interested in applying empirical rational tools to nearly every domain of life.
How to sleep better. How to have better skin. How to parent. How to prevent colds. How to pick college courses. How to be good at sex. How to find a partner. How to use data to decide who to marry. How to think. How to make your room as bright as the outdoors. How to read self-help. How to optimize. How not to optimize — and what to optimize for. How to solve problems in general.
This sometimes accumulates to a genre of writing I love just as much:
There has never been a better time to have a problem, and if you think there is a problem you cannot solve, maybe you’re not actually trying.
There seems to be an important distinction to be made about general life optimization, which I describe as hacks, heuristics, and frameworks. A hack is a solution to one problem; a heuristic is, in some ways, a reusable hack; and a framework is something that tells you which problems are worth solving and why.
Why should we try to optimize sleep? Because better sleep leads to better subjective experience and better performance. And in general, better performance is desirable, and better subjective experience is good, or something like that. Why should we date? Maybe because social relationships almost always make people happier, maybe also because dating is conducive to marriage, and marriage is a stable institution suitable for raising children. Why should we raise children? Maybe because demographic collapse is a real problem, maybe there are selfish reasons to have more kids, and maybe you don’t want any of these things but it is good to have options, and ways to go about it when you do.
Hacks don’t replace frameworks. They smuggle in implicit frameworks while pretending to be framework-free. These “super basic notions that we have all internalized as obviously good” are residues of intellectual traditions. “Health is good” is based on the modern affirmation of ordinary biological life; stoics were indifferent to bodily states and medieval Christians valued mortification of the flesh. “Time is a finite resource and saving it is good” draws on a Protestant-capitalist conception of time as something to be spent wisely, which would have been incomprehensible to most pre-modern cultures with cyclical time. “ Individual agency over your life is possible and desirable” relates to Enlightenment autonomy and Romantic authenticity. Charles Taylor calls these notions constitutive goods, moral sources that power our evaluations that we’ve so thoroughly internalized them that we take them as neutral facts about the world.
It is important to note that the genealogical observation (these values have specific historical origins) and the metaethical claim (some moral truths are objectively correct) are not necessarily in conflict — but the relationship between them is complicated. The tension isn’t between genealogy and objective morality. It’s between the first-order moral claims (health is good, suffering is bad — these might well be objective) and the ordering of those claims relative to each other, the absence of any shared, explicit ordering — even if the individual values being ordered might be correct.
Since they can no longer combine in the service of a supreme value, [the value-systems] claim equality one with the other: like strangers they exist side by side, an economic value-system of 'good business' next to an aesthetic one of l'art pour l'art, a military code of values side by side with a technical or an athletic, each autonomous, each 'in and for itself,' each 'unfettered' in its autonomy, each resolved to push home with radical thoroughness the final conclusions of its logic and to break its own record.
— Hermann Brooch, The Sleepwalkers
The modern secular person has implicit frameworks (eg. human rights, individual dignity, the value of flourishing), plus an enormous and growing stack of hacks built on top of it, but has lost explicit frameworks that would organize these implicit goods into a hierarchy.
If all genuine goods are harmonious, you don’t need a framework — you just need more hacks. Optimize health and productivity and relationships and meaning and agency, and they’ll all reinforce each other. The hack ecosystem implicitly assumes this: every “10 things to improve your life” list treats its items as additive, or as inspirational ideas you can freely adopt or ignore. There’s a related notion that hacks are just random things that might or might not work for you, and you should simply try them and keep what sticks. But if goodness does compete — if real goods genuinely conflict, and pursuing one requires sacrificing another — then hacks are structurally incapable of helping you at the decisions that matter most. And it is obviously true that you can neither implement all 50 suggestions to your life, or pick all options between every decision. If we put such effort into solving object-level problems — optimizing sleep, nootropics, workflows — it seems odd to make value-level decisions arbitrarily.
This is probably also why the EA framework resonates with people even when they might disagree with its conclusions. It at least attempts an ordering. Neglected causes matter more than popular ones. Expected value calculations can guide your choices when goods conflict. Et cetera. You can argue with any of those claims, but at least there are claims to argue with.
Ulrich took it as a matter of course that a man who has intellect has all kinds of intellect, so that intellect is more original than qualities. He himself was a man of many contradictions and supposed that all the qualities that have ever manifested themselves in human beings lie close together in every man’s mind, if he has a mind at all. This may not be quite right, but what we know about the origin of good and evil suggests that while everyone has a mind of a certain size, he can still probably wear a great variety of clothing in that size, if fate so determines.
And so Ulrich felt that what he had just thought was not entirely without significance. For if, in the course of time, commonplace and impersonal ideas are automatically reinforced while unusual ideas fade away, so that almost everyone, with a mechanical certainty, is bound to become increasingly mediocre, this explains why, despite the thousandfold possibilities available to everyone, the average human being is in fact average. And it also explains why even among those privileged persons who make a place for themselves and achieve recognition there will be found a certain mixture of about 51 percent depth and 49 percent shallowness, which is the most successful of all. Ulrich had perceived this for a long time as so intricately senseless and unbearably sad that he would have gladly gone on thinking about it.
— Robert Musil, The Man Without Qualities
Part of the enthusiasm about agency emerges from a concern that people don’t really know what their future looks like, and they desire to control or lay claim to it in a way they hope agency will provide. In other words, the agency trend might itself be a symptom of framework loss. When we run out of answers to “what should I do with my life?”, the next best thing is to get really good at doing things in general.
Man ... is driven out into the horror of the infinite, and, no matter how he shudders at the prospect, no matter how romantically and sentimentally he may yearn to return to the fold of faith, he is helplessly caught in the mechanism of the autonomous value-systems, and can do nothing but submit himself to the particular value that has become his profession, he can do nothing but become a function of that value — a specialist, eaten up by the radical logic of the value into whose jaws he has fallen.
— Hermann Brooch, The Sleepwalkers
I’m not sure what can serve as a genuine framework. This seems like one of the most difficult problems imaginable. But I am confident that there exist a failure more of letting hacks or heuristics harden into pseudo-frameworks. We take something that is actually an empirical claim bundled with a value judgment and compress it into a theoretical principle that feels like it can organize everything. “Passions are malleable” “you can just do things” or “government power tends to expand, so be cautious” can itself become a mind virus if you start treating it as a universal principle rather than a contextual heuristic. When a contextual truth gets promoted to a universal ordering principle and becomes load-bearing for your entire worldview, challenging it feels like pulling a thread that might unravel everything.
It is probably worse to filled the framework-shaped hole with something that doesn’t actually fit, but that sits comfortably enough to make the searching feel unnecessary; than admitting to having no framework at all.
2026-03-28 10:59:56
There's a question that's held my fascination for months. At times, it's had me spinning in circles, caught up in seemingly impossible contradictions. And it's not the famous Sleeping Beauty problem... or at least, not exactly.
There is a certain Vincent Conitzer, of Duke University, who presents what he calls a "Devastating" example supposedly disproving the logic behind the "1/2" answer to the original problem.
His argument is valid, but based on a faulty premise. He assumes that Halfers, to get their answer, always look at remaining possibilities and renormalize equally between them. (At no point does he justify this assumption.)
So Conitzer is making a mistake in his critique of the Halfer position. I would say that anyone who still believes in the "1/3" answer to the original SB problem is making a mistake. But I'm not any less error prone myself. Holy hells, have I made mistakes. Conitzer's example is a variant of SB with an easier answer, because it's a question where the correct answer and the intuitive answer are the same. Yet at one point I got so confused, I believed in an answer that really should be self-evidently wrong to anyone.
The question that's fascinated me for months isn't any particular object-level SB variant, but rather the meta-level question: What makes Sleeping Beauty so difficult?
I think the typical answer here is, in a word, "anthropics".
Take, for instance, Eliezer Yudkowsky, who in Project Lawful writes (via a character he uses to serve as a mouthpiece for lecturing on rationality):
This however would get us into ‘anthropics’, and we are not getting into ‘anthropics’. That, by the way, is a general slogan of dath ilani classes on probability theory: We are not getting into ‘anthropics’. I’m not even going to translate the word. As long as you don’t make any copies of people, you can stay out of that kind of trouble. That trouble-free life should be our ambition for at least the next several weeks.
I’m not satisfied with that. “Anthropics” doesn’t get a free pass to be inexplicably mysterious. It should bend to the laws of math, like everything else.
This is a post with a two-part thesis:
But before we dive into the specifics of Sleeping Beauty and its variants, first I'd like to start with a very different example with surprisingly similar underlying math...
“Well, a remarkable thing about this problem, simple as it is, is that it has sparked just endless debate”
In the Monty Hall problem, when you first pick a door, the chance of it being the right one is 1/3. When Monty opens a door and then you choose to switch, you’ve now narrowed down a possibility space from 3 options to 2: It can either be behind door 1 or door 3. A 50/50 chance! Yay! That’s an improvement.
Well, actually, that’s wrong. It’s not 50/50. If you switch, your chances go up to ⅔.
According to Conitzer, the “Halfer rule” says that whenever possibilities are eliminated, you should renormalize between all remaining possibilities equally. Here, that gets the wrong answer of 2/3.
I should hope no one uses Conitzer's inaptly-named "Halfer rule". Let’s see instead how my proposed solution of relying on Bayes handles this problem:
Say that you guess door 1, Monty opens door 2 to reveal a zonk, and you decide to switch to door 3.
Before any choices have been made, our prior probabilities for each door hiding the sports car are as follows:
P(1) = P(2) = P(3) = 1/3
The prior probability it’s not behind door 2?
P(~2) = ⅔
We want to know the chance it’s behind door 3, given that it’s not behind door 2, AKA:
P(3 | ~2) = ?
We also know that if it’s behind door 3, it cannot be behind door 2:
P(~2 | 3) = 1
All that’s left is Bayes:
P(3 | ~2) = P(~2 | 3)*P(3)/P(~2) = (1)*(1/3)/(2/3) = 1/2
Somehow, we got the wrong answer of 1/2 again.
But that's alright. We'll come back to this.
Conitzer’s variant begins like this: Sleeping Beauty gets put to sleep, wakes up Monday morning, and is shown a coin flip toss. Then regardless of the coin flip’s result, her memory of Monday will be erased and the procedure repeated: She gets put to sleep, wakes up Tuesday, is shown a second coin toss, and has her memory again wiped.
Conitzer’s question: After waking and observing a coin flip to Heads, what should Beauty believe is the chance that the two coin flips will have had different results? I.e., what’s the chance that either today is Monday and the coins will be Heads-Tails (HT) OR today is Tuesday and the coins were Tails-Heads (TH)?
If you believe that, after observing Heads, you should update to P(coins are different) = 2/3, then consider that the situation is symmetrical with Tails. Upon observing Tails, you should also update to 2/3. But that means no matter what you observe, you will update to 2/3—and if you already know this update will happen, you should just go ahead and update now. But that would mean believing two fair coins have a 2/3 likelihood of flipping to different results, even before having observed anything.
(For a period of time, this is what I actually believed! The reason for my confusion wasn't necessary to the main points of this post, but if you're interested, I explain in the bonus section at the end.)
Again, the “Halfer rule” (of renormalizing equally among remaining possibilities) gets the wrong answer of ⅔. That logic goes like this: When you observe a Heads, you learn that Tails-Tails (TT) has not happened. Because you don’t know which day it is, however, you haven’t learned anything else. That means all of HH, HT and TH remain as possibilities…
…with equal likelihood. HT and TH are two of the three options, so the chance the coin flips are different must be two out of three.
What about Bayes?
P(Coins are different | observe ~TT) = ?
P(Different) = 1/2
P(~TT) = 3/4
P(~TT | Different) = 1
P(Different | ~TT) = P(~TT | Different) * P(Different) / P(~TT) = (1)*(1/2)/(3/4) = 2/3
These get the same results because they mean the same thing. Eliminate TT, and this is what happens.
This is a fun problem! It’s weird. Unless you already understand it perfectly, you should be surprised by this. If you feel confused, that’s a good thing! (And if you're not confused, then that's also a good thing, and kudos.)
We woke up, we observed a Heads, we logically eliminated TT from the possibilities, and of course could not eliminate any of HT, TH, or HH. So where’s the flaw in the above reasoning?
There’s a different Bayes calculation we can apply to the Monty Hall problem that will give us the correct answer:
P(observe ~2) = 1/2
P(~2 | 3) = 1
P(3 | ~2) = P(~2 | 3)*P(3)/P(~2) = (1)*(1/3)/(1/2) = ⅔
Except this only works because I redefined the meaning of “~2”. Before, it referred to the initial chance that Door 2 would be hiding the car. Now, it means: Once you’ve chosen to open Door 1, what’s the chance that Monty will reveal a zonk by opening Door 2?
He has to choose either Door 2 or Door 3 to open, and they’re equally probable from our perspective, hence: P(~2) = 1/2. If the car is behind Door 3, then we know Door 2 must be opened, and thus: P(~2 | 3) = 1. Then the rest is just plug-and-play with Bayes.
Our earlier calculation wasn’t wrong. It just didn’t capture as much information about the situation. True and accurate statements can still be incomplete.
But…
…you might have noticed…
…this doesn’t actually resolve anything. If both mathematical calculations are accurate, how do we choose which one to use over the other?
The answer: You don’t have to choose one over the other. You can just use both! If you’ve got two pieces of evidence, A and B, then Bayes lets you update on A first, then followed by B. Or B followed by A. That’s true even if A happens to be a superset of the information with B. No matter how you slice it, the math will work out.
A penguin or a cow? It’s both! Source: This wonderful visual anagrams gallery
Let “2” mean the event that Door 2 is hiding the car.
Let “O2” mean the event that Monty will Open 2.
Our initial calculation showed:
P(3 | ~2) = 1/2
Now we want to update on O2, in addition to just ~2:
P(3 | observe O2 ^ ~2) = ?
If door 2 doesn’t have the car but door 3 does, Monty will be forced to open door 2:
P(O2 | 3 ^ ~2) = 1
If you only know that the car’s not behind door 2, then there’s a 1/2 chance it’s behind door 3 and 1/2 it’s behind door 1:
P(O2 | ~2) = (1/2)(1) + (1/2)(1/2) = 3/4
Then applying Bayes:
P(3 | O2 ^ ~2) = P(O2 | 3 ^ ~2) * P(3 | ~2) / P(O2 | ~2) = (1)(1/2)/(3/4) = 2/3
Now let’s try going in the opposite direction, starting with the update we calculated using event O2, and seeing what happens when add in “2”:
P(3 | O2) = 2/3
P(3 | observe ~2 ^ O2) = ?
This one’s easy:
P(~2 | O2) = 1 (if Monty opens door 2, door 2 can’t have the car)
P(~2 | 3 ^ O2) = 1 (ditto)
P(3 | ~2 ^ O2) = P(~2 | 3 ^ O2) * P(3 | O2) / P(~2 | O2) = (1)(⅔)/(1) = 2/3
O2 implies ~2, so we get no change when considering ~2 as a subsequent, standalone update.
Earlier, I considered the event “TT” and described:
P(~TT) = 3/4
P(~TT | Different) = 1
Let’s instead consider the observation of not whether you will, on either Monday or Tuesday, eventually observe a Heads, but consider the event of observing a Heads right now, in this particular moment of observation, and let’s call that event “H”.
P(H) = 1/2
Because, you know… fair coin. But here’s a diagram anyway:
P(H | Different) = 1/2
Because if there’s one Heads and one Tails, but you don’t know which will be on which day, and you don’t know which day it is, then yeah… it’s still an even chance.
And now the calculation:
P(different | H) = P(H | different) * P(different) / P(H) = (1/2)*(1/2)/(1/2) = 1/2
Like with the Monty Hall problem, we have two ways of looking at the situation and two similarly defined events, one which gets the right answer (in this case, “H”) and one that does not (“~TT”). It can seem like a paradox: Without already knowing the right answer, how do we know to use “H” instead of “~TT”? And as before, if you’re not sure about which event to update on, you can simply update on both.
Let’s confirm.
If we wake up and see Heads, then we know Tails-Tails isn’t happening. Therefore H → ~TT, and we should expect that if we update on H first, a subsequent update on ~TT should effect no change:
P(Different coins | observe ~TT ^ H) = ?
P(~TT | different ^ H) = 1
P(~TT | H) = 1
P(different | H) = 1/2 from Step 1
P(different | ~TT ^ H) = P(~TT | different ^ H) * P(different | H) / P(~TT | H) = (1)*(1/2)/(1) = 1/2
And in the other direction, if we update on ~TT first and then H?
Calculated previously:
P(different | ~TT) = ⅔
Also, if we know Tails-Tails didn’t happen, then we know the chance of experiencing a Heads waking is higher:
P(H | ~TT) = ⅔
Also, if we know the coins are different, then we already know ~TT. Since we already established that P(H | different) = 1/2, that means this is the same:
P(H | different ^ ~TT) = 1/2
Putting it all together:
P(Different | H ^ ~TT) = P(H | different ^ ~TT) * P(different | ~TT) / P(H | ~TT) = (1/2)*(2/3)/(2/3) = 1/2
I love this kind of thing, math working out the way you expect it to.
So the general guidance I’d like to give, if you’re seeing Bayes spit out different answers and you’re not sure which to go with: Just go with both! Add as much information as you can assemble, then see where the math takes you.
Furthermore, I think there will always be a clue you can find in retrospect which will act to confirm your findings. For instance:
“The Equiprobability Bias: [Possible] Events tend to be viewed as equally likely” — Joan B. Garfield
Roll two dice, and you can get any sum between 2 and 12. These sums are not all equally likely; there are more ways to sum to 7, for example, than to get snake eyes. Yet “people show a strong tendency to believe that 11 and 12 are equally likely” (Nicolas Gauvrit and Kinga Morsanyi).
A better version of this dice problem was made famous by the Grand Duke of Tuscany and Galileo, around the year 1620. The Grand Duke considered three dice rolls, then asked Galileo if the sum of “10” was any more likely than the sum of “9”.
As Florent Buisson explains here, both sums can be achieved in six ways, via the following permutations:
9 = 6 + 2 + 1 = 5 + 2 + 2 = 5 + 3 + 1 = 4 + 3 + 2 = 4 + 4 + 1 = 3 + 3 + 3
10 = 6 + 2 + 2 = 6 + 3 + 1 = 5 + 3 + 2 = 5 + 4 + 1 = 4 + 4 + 2 = 4 + 3 + 3
It’s easy to think that “9” and “10” must be equally probable, and yet: “10” is more likely than “9”, because though it has the same number of summing permutations, “10” has more combinations. (4+3+3 is three times as likely to result as 3+3+3, which can only result if all dice rolled a 3.)
We saw the exact same dynamic at play with Monty Hall, thinking that the two doors that remain after one is eliminated must each be 1/2.
The same thing happens again with Bertrand’s box paradox.
Are you surprised by Benford’s Law? If so, it counts towards the Equiprobability Bias as well.
It should make sense: When we first learn probability as kids, we’re trained on examples involving coins and dice and drawing cards from decks, all equiprobable events. It’s like the $1.10 ball-and-bat problem: a failure of over-generalizing simple rules and not working through the necessary steps.
With SB, we’re being presented three equal-seeming options. Then we over-generalize. I’m no exception to this—I used to be a Thirder myself, until I thought through the problem more. And the gods know I’ve made all sorts of reasoning and mathematical errors in my life.
But I also think there are some very smart people getting unnecessarily tripped up by SB due to a tendency to get tangled up into webs as they contend with the aforementioned topic of anthropics.
There’s another problem I’ve yet to mention in this post, despite the fact that it is completely isomorphic to Conitzer’s SB variant.
And by “isomorphic” I mean the math is EXACTLY the same.
It’s called the “Boy or Girl paradox”, and I don’t think it should be any more or less difficult to understand than Conitzer’s problem.
Except it is.
I don’t think Bentham’s Bulldog and other Thirders would trip up on the Boy or Girl paradox in the same way. Where the Boy or Girl problem merely involves the meeting of kids or dogs in different contexts, Sleeping Beauty uses amnesia to create experientially identical “observer moments”. Thinking about observer moments, or anthropics, can lead you down twisty chains of thought.
For instance: In the Conitzer variant, some would argue that the difference between events “H” and “~TT” is one of “self-locality”: ~TT is a description of the world from a more removed, objective perspective. H is a description of you during a particular observer moment. This is the fundamental difference, they’d argue. But this sort of thinking gives rise to paradoxes, which then require all sorts of convoluted concepts like the “SIA” or “SSA” to resolve, and can lead to EXTREMELY absurd conclusions such as the presumptuous philosopher.
As I argue here and Ape in the Coat argues here, neither the SIA nor the SSA is correct because both rely on the Doomsday argument’s faulty anthropic assumptions.
We don’t need “self-locality” when we can simply recognize that “H” holds more informational content than “~TT”.
Except I’m not satisfied leaving it at that.
How is it I was once a Thirder myself, and yet I’ve never believed that the 1/2 Boy or Girl problem could have any other answer than 1/2? If the math is the same, where does the extra confusion come from?
I believe I’ve found my answer, and it’s the same answer I’ve already presented: The Equiprobability Bias.
That’s really it, I think: the seed from which all these other tangled roots have grown.
If you have any other SB variants you’ve struggled with, or that you’ve created, please share them! I bet that with Bayes, we can find their answers.
Let’s say that during Conitzer’s setup, after Beauty awakes and sees a Heads, she meets a bettor who explains:
I decided to pick a random “Heads” day to approach you.
(This means if TT happened, I would not have approached.)
Do you want to take the bet that both coins flipped Heads?
If you’re right, you’ll gain $3. Wrong and you’ll lose $2.
Previously, I showed that P(Different | observation) = 1/2. If that’s the case, then the expected payoff for accepting the bet would be (1/2)(+$3) + (1/2)(-$2), which is positive. You should accept the bet!
Right?
Well, actually…
The expected payoff is (1/3)(+$3) + (2/3)(-$2), a negative value, which makes this a bad bet.
It seems as if there’s a sort of logic that’s needed to get the right answer to Conitzer’s problem, which leads to 1/2, and there’s another sort of logic needed to get the right answer here, which leads to 2/3. We distinguished them with the descriptors of “Beauty will at some point see Heads” versus “Beauty is currently seeing Heads”—a more general statement versus a “self-locating” one.
That’s what a particular r/slatestarcodex redditor argued with me, and it tripped me up bad. We went in circles and circles on this, yet at no point did either one of us realize that those 2/3 values represent different things.
This becomes easy to see if we consider one last variant, wherein the bettor approaches Beauty but changes the conditions of his approach:
I chose a random day (Heads or Tails) to approach you.
Do you want to take the bet (same as before)?
In this case: Yes! Take the bet! The chance the coins are different or the same really is 1/2 and you stand to gain in expected value.
The key is this: “I decided to pick a random “Heads” day to approach you” is new information. “Beauty is currently seeing Heads” represented the most up-to-date information before the bet, with its answer of 1/2. Learning about the approach creates an update towards 2/3…
…and makes this yet another example of the guideline: When in doubt, just try Bayes again.
