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Has some government or random billionaire sought out Petrov's heirs and made sure none of them have to work again if they don't want to? It seems like an obviously sensible thing to do from a game-theoretic point of view.
Not at the scale you're suggesting, but relevant: https://futureoflife.org/recent-news/50000-award-to-stanislav-petrov-for-helping-avert-wwiii-but-us-denies-visa/
It seems like an obviously sensible thing to do from a game-theoretic point of view.
Hmm, seems highly contingent on how well-known the gift would be? And even if potential future Petrovs are vaguely aware that this happened to Petrov's heirs, it's not clear that it would be an important factor when they make key decisions, if anything it would probably feel pretty speculative/distant as a possible positive consequence of doing the right thing. Especially if those future decisions are not directly analogous to Petrov's, such that it's not clear whether it's the same category. But yeah, mainly I just suspect this type of thing to not get enough attention that it ends up shifting important decisions in the future? Interesting idea, though -- upvoted.
. . . Is there a way a random punter could kick in, say, $100k towards Elon's bid? Either they end up spending $100k on shares valued at somewhere between $100k and $150k; or, more likely, they make the seizure of OpenAI $100k harder at no cost to themselves.
That's one millionth of the bid, 0.0001%. I expect the hassle of the paperwork to handle there being more than one bidder to be more trouble than it's worth, akin to declaring a dollar you picked up on the street on your income tax forms.
True. But if things were opened up this way, realistically more than one person would want to get in on it. (Enough to cover an entire percentage point of the bid? I have no idea.)
You want to be an insignificant, and probably totally illiquid, junior partner in a venture with Elon Musk, and you think you could realize value out of the shares? In a venture whose long-term "upside" depends on it collecting money from ownership of AGI/ASI? In a world potentially made unrecognizable by said AGI/ASI?
All of that seems... unduly optimistic.
I once saw an advert claiming that a pregnancy test was “over 99% accurate”. This inspired me to invent an only-slightly-worse pregnancy test, which is over 98% accurate. My invention is a rock with “NOT PREGNANT” scrawled on it: when applied to a randomly selected human being, it is right more than 98% of the time. It is also cheap, non-invasive, endlessly reusable, perfectly consistent, immediately effective and impossible to apply incorrectly; this massive improvement in cost and convenience is obviously worth the ~1% decrease in accuracy.
8I think they meant over 99% when used on a non-randomly selected human who's bothering to take a pregnancy test. Your rock would run maybe 70% or so on that application.
This is a general problem with the measure of accuracy. In binary classification, with two events and , "accuracy" is broadly defined as the probability of the "if and only if" biconditional, . Which is equivalent to . It's the probability of both events having the same truth value, of either both being true or both being false.
In terms of diagnostic testing it is the probability of the test being positive if and only if the tested condition (e.g. pregnancy) is present.
The problem with this is that the number is strongly dependent on the base rates. If pregnancy is rare, say it has a base rate of 2%, the accuracy of the rock test (which always says "not pregnant", i.e. is always negative) is
Two better measures are Pearson/Phi correlation (which ranges from -1 to +1), and the odds ratio, which ranges from 0 to , but which can also be scaled to the range [-1, +1] and is then called Yule's Y.
Both correlation and Yule's Y are 0 when the two events are statistically independent, but they differ for when they assume their maximum and minimum values. Correlation is 1 if both events always co-occur (imply each other), and -1 if they never co-occur (each event implies the negation of the other). Yule's Y is 1 if at least one event implies the other, or the negation of at least one implies the negation of the other. It is -1 if at least one event implies the negation of the other, or negation of at least one event implies the other.
This also means that correlation is still dependent on the base rates (e.g. marginal probability of the test being positive, or of someone being pregnant) because the measure can only be maximal if both events have equal base rates (marginal probability), or minimal if the base rate of one event is equal to the base rate of the negation of the other. This is not the case for odds ratio / Yule's Y. It is purely a measure of statistical dependence. Another interesting fact: The correlation of two events is exactly equal to Yule's Y if both events have base rates of 50%.
1There's an upside of conventional education which no-one on any side of any debate ever seems to bring up, but which was a major benefit (possibly the major benefit) of my post-primary studies. Namely: it lets students discover what they have a natural aptitude for (or lack thereof) relative to a representative peer group. The most valuable things I learned in my Engineering courses at university were:
.I'm pretty mediocre at Engineering, especially sub-subjects which aren't strictly Structural and/or Mechanical.
.In particular, I'm significantly worse than the average would-be Engineer at working with electronics, so I should change my plans to specialize in that field.
.Conversely, I'm significantly better than the average would-be Engineer at work involving code, money, probability, simulation and inference. (Learning this is a large part of why I eventually left Engineering for Data Science and Finance.)
Findings like this aren't perfect since they can give false negatives (a poorly-taught course can lead someone to conclude they don't like the subject when they actually don't like the teacher) and false positives (my initial showing at coding courses made me think I had a genius-level talent for programming; turns out I actually just have a genius-level talent for beginner programming, and much of my lead evaporates when doing anything nontrivial); also, there's no hard reason they can't also be found in the workforce (if people joined companies at the same time as two dozen similarly-aged and -educated peers with whom they were frequently compared, they'd get this benefit while being paid[1]). Still, I think it's an undervalued and underdiscussed output of our current educational paradigm.
- ^
Soldiers get this, but I'm not thinking of that kind of company.
Lower switching costs when you're in the middle of a degree, maybe? You can just take courses in a closely related domain, or work as an assistant in a different lab, in a much more fluid and straightforward manner, versus having to apply to a different job and get through the interviews and pay a significant upfront cost before you even get to the nuts and bolts of stuff.
1This aligns with my take that it's actually fine for education's primary purpose is as a way to signal to companies cheaper and more reliably, even if teachers can't do anything because way more ability is genetic than people like to admit in some corners of polite society, and this combined with actually imparting social skills is probably the main benefit of education.
Bryan Caplan argues that we shouldn't subsidize such signaling mechanisms by governments.
Edited to not mischaracterize Bryan Caplan's view.
he doesn't really see any value in systems that are there purely to signal.
This seems like a mischaracterization of his view. I’m pretty sure he thinks its wrong to subsidize such signaling mechanisms.
First off signaling is relative, so if (say) everyone goes to high school and only the very best go to college, from a signaling perspective, this is just as useful a signal as everyone going to college and only the very best go to grad school. Therefore we should not spend public dollars getting more people to go to college.
Second, in the signaling framework, there are no externalities to schooling kids, so there is no market failing to correct with (say) the government subsidizing the debts of college students.
Third, due to the first point, if any major market failure is present its the tendency to get into signaling spirals, where the positive signal of (say) a high school education degrades over time, making everyone spend more years and dollars in college getting what was once the same signal as a high school diploma. More years of schooling here is a cost, which everyone would prefer to pay less of. So insofar as there’s any case for government involvement it ought to be a tax, not a subsidy.
"What important truth do you believe, which most people don't?"
"I don't think I possess any rare important truths."
How could we test the inverse? How do we test if others believe in rare important truths? Because obviously if they are rare, then that implies that either we don't share them, therefore do not believe they are truthful or important.
"Mel believes in the Law of Attraction, he believes it is very important even though it's a load of hooey"
I suppose there are "Known-Unknowns" and things which we know are significant but kept secret (i.e. Google Pagerank Algorithm, in 2008 the 'appetite' for debt in European Bond Markets was a very important belief and those who believed the right level avoided disaster), we believe there is something to believe, but don't know what the sin-qua-non belief is.
I recently watched (the 1997 movie version of) Twelve Angry Men, and found it fascinating from a Bayesian / confusion-noticing perspective.
My (spoilery) notes (cw death, suspicion, violence etc):
- The existence of other knives of the same kind as the murder weapon is almost perfectly useless as evidence. The fact that the knife used was identical to the one the accused owned, and was used to kill so close to when the defendant's knife (supposedly) went missing, is still too much of a coincidence to ignore. The only way it would realistically be a different knife is if someone was actively trying to frame the defendant, and arranged for his knife to be lost at the same time; and if they could do both of those things, it makes more sense for Hypothetical Secret Mastermind X to just stab the victim with the accused's actual knife. (This means Juror 8's illegal purchase of an identical knife in the name of justice was epistemically pointless, and only served to muddy the waters; I'm oddly enamored by the probably-accidental pro-Lawful-Good thematic implications.)
- The old man's testimony is suspect for more reasons than the jurors notice. The lack of fingerprints on the murder weapon suggests the culprit wiped it off first, but the old man claims the culprit ran off immediately after the body hit the floor. However, this aligns with the other reason to consider him unreliable, which is him (allegedly) managing to move quickly enough to see the accused leave the scene; it seems pretty plausible that he got the timing wrong but everything else right.
- The paramedic juror's claim that the knife was used incorrectly - that it's the kind of knife made to stab up through the gut instead of down through the ribs - doesn't exonerate the defendant, and might actually incriminate him. It's a fact about the knife, not the user; if anything, a young man might be more likely than the average assailant to wield his weapon wrong.
- The other witness turning out to (probably) habitually wear glasses doesn't necessarily make her testimony invalid. She could be farsighted, could need reading glasses, or could just habitually wear them to seem intelligent or as a fashion statement. All of these explanations seem more likely than a - by all accounts, scarily competent - prosecutor putting her on the stand without checking she could actually see the murder. (None of the jurors consider requesting additional testimony on this topic, even though it's both easy to check and the point which ends up deciding the final verdict.)
From all the above, I conclude:
The accused is very likely to have committed the murder.
and
The protagonist probably has some kind of agenda: either he takes issue with capital punishment, knows the defendant personally, strongly dislikes the carceral justice system, is being bribed, or is trying to arrange acquittal for a guilty party just to see if he can.
However
I still think a case can be made for the existence of reasonable doubt.
if and only if
You consider the possibility it was a suicide.
(trigger warning for detailed discussion of that thing I just mentioned)
If I knew for a fact the defendant was innocent, most of my probability mass would be on some variation of the following sequence of events.
- The 'victim' has his (injury-free) altercation with the accused. This rattles the accused to the point that he forgets to take his knife with him when he leaves for the movies; he falsely assumes that it "fell out of his pocket".
- The 'victim' is also rattled, and decides to commit suicide. (Possible motivations: realizing that he can no longer reliably win a fight against the target of his abuse and wanting to quit while he's ahead, feeling regret about his treatment of the accused, being angry at the accused and wanting to die in such a way that the accused ends up accused.)
- The 'victim' stabs himself in the chest, and not through the gut, in an attempt to end his life as quickly, painlessly, and dramatically as possible. Possibly he shouts "I'm going to kill you!" as he does this, either out of genuine self-loathing or an attempt to implicate the accused; possibly he makes a point of staggering around near an open window with a knife sticking out of his chest before collapsing; alternatively, the witness testimonies may just be mistaken and/or falsified for reasons discussed in the film.
- The accused returns home to find a dead body and two policemen. Between the lingering effects of earlier events, the presence of the victim's corpse, his current predicament and (quite plausibly) some mind-altering substances he chooses not to admit to using in the subsequent trial . . . the accused finds himself unable to provide satisfactory answers when the police ask for the titles and lead actors of the movies he watched. (He may or may not be able to recall other details about these movies: but either he doesn't think to volunteer this information and the police don't ask for it, or the police choose not to record inconvenient facts in an attempt to close the case cleanly while technically telling the truth.)
This hypothesis makes sense of the paramedic's claim about the type of knife, makes sense of the silent evidence of neither the accused nor the corpse having any injuries mentioned aside from the single stab wound (a person comfortable with violence yells an explicit verbal warning at another person comfortable with violence, and then stabs him to death, but there's no sign of a struggle?), and is supported by base rates (suicide is significantly more common than homicide in first-world nations).
. . . to be clear, I'd still say murder is much more likely, but I consider the above possibility just possible enough to be conflicted about the reasonableness of reasonable doubt in this case.
I'm curious what other LW users think.
One last, even more speculative thought:
Literally everything the racist juror does in the back half of the movie is weird and suspicious. It's strange that he expects people to be convinced by his bigoted tirade; it's also strangely convenient that he's willing to vote not guilty by the end even though he A) hasn't changed his mind and B) knows a hung jury would probably eventually lead to the death of the accused, which he wants.
I don't think it's likely, but I'd put maybe a ~1% probability on . . .
. . . him being in league with the protagonist, and them running a two-man con on the other ten jurors to get the unanimous verdict they want.
Mostly here because I was active a long time ago, but this is interesting enough to make an account again. If I use language that doesn't fit the lingo, that's why.
So you know if you want to bother reading-My conclusion is that I'm pretty sure you're wrong in ways that are fun and useful to discuss!
First off, it's absolutely relevant that the accused's knife isn't unique. If the knife it unique it selects for them specifically out of the suspect pool of everyone-it's not perfect evidence, it's possible that they lost it, but in most worlds where it's a unique knife they're the one who uses it on someone.
Following that, and this is contextual to the story and setting, the prior probability of someone owning a knife in his community (New York young man from a vaguely Hispanic community) is high. If the specific knife is thus common enough that it's possible to find a copy within a block or so, the chance that any suspect would have access to that knife becomes pretty reasonable. I'd vaguely estimate we're supposed to estimate a greater than 90% probably that any kid of his ethnicity and gender owns a knife, and at that point we're looking at hundreds of suspects who we know could have committed the crime with a perfect copy of the murder weapon because we know one was within walking distance. We know at least two of those knives existed near the murder, so let's spitball that the chance of anyone else using that knife is about equal to the chance of the suspect using the knife. 50-50. It's much more likely that the suspect had his knife than anyone else, but there are so many other people who could have committed the crime that I don't feel comfortable saying he's much more likely than everyone.
(In reality the ownership rate of knives was likely lower among Hispanic teens in 1950's America, the media exaggerated it, but people might borrow murder weapons, so we're into speculation land).
Second, the flaws in the eyewitness testimony are sufficient to basically completely obviate any of the witnesses as interrogators of truth. Eyewitnesses in perfect health and alacrity make major mistakes. There's too much evidence to summarize, but we're talking on the order of 75% chance of correct identification at absolute best. Arguably eyewitness testimony has basically no impact on probability estimates before we start adding in reasons to doubt it. In this context the stated circumstances-that the witness is elderly and saw the murder in the dark through a train-are actually sufficient to make it highly unreliable before we talk about glasses, which are actually just a red herring. Eyewitness testimony is so bad that they should have rejected it before they learned the witness even might be blind! If I was forced to give a prior probably of how likely that eyewitness was likely to pick the right suspect out of a lineup of all suspects in those circumstances, it'd be on the order of one in a thousand. There are likely nearly that many young people of vaguely correct build in a few block radius, and I have nearly zero confidence they could pick one out from another in the stated circumstance. I'm being generous by several orders of magnitude if they have vision problems.
Eyewitness testimony is that awful. It's the source of most miscarriages of justice, and it's arguably simply a mistake to use it in murder trials at all-the correct move is probably to ban it because it's more prejudicial to juries than evidential, even if a glimmer of scientific validity can be gleamed from it by an expert.
Hence the two/three pieces of reasonable evidence are-Knife, and fight with victim (whom was his dad). The fight is not great evidence, but specific threats to family members precede a lot of murders, believe it or not.
The prior probably that kid is guilty because the victim was his dad is around 10% (yes, really), the fight and lack of other familial suspects might increase that to 20%, and as I said I don't consider the knife as raising the odds more than a coin toss. The eyewitness testimony is a rounding error.
I'd hence conservatively put the kids chance of being guilty at ~70%. And that's completely unacceptable for a murder trial. That's barely a preponderance of evidence.
Legally, the correct move at this point is for the jury to walk out and admit everything they've discussed openly, at which point the entire case will be thrown out for a mistrial. Technically Juror 12 could have forced that himself by just saying that he saw an identical knife on the street, which is only a criminal (well...highly legally dubious) act when he smuggles it into the room. That's not good for the boy, because the idea that only one man in twelve was willing to murder him through legal maleficence for being Hispanic in 1950's new York is laughable, but it's the legal option. Juror 12, knowing that, is probably morally justified in pushing for an acquittal anyway.
As for shaming the racist juror into silence-shame is a powerful motivator. Racists, particularly modern racists, are often fundamentally social and physical cowards too, and they just bowled over him (mostly illegally) rather than relying on him to actually admit fault and chance his vote, which is more realistic to the psychology involved. Also fully justified, of course.
We can likely reject your alternative hypothesis because of the prior probably that someone who wanted to rig a trial could do so. There are tens of thousands of people whom might be called for jury duty (accounting for racial and gender discrimination...) on a given day in 1950's New York. Even knowing the day, hundreds are then called, of which 14 get put on a 12 man jury with two substitutes. You can go find the real numbers to do the math if you want, but back of the envelope, without a way to rig that system you're looking at a chance of less than 1/10000 if you knew that the juror wanted to rig the trial ahead of time. Rigging the system to influence juror selection is rare enough to have an infinitesimal prior probability if you know someone wants to. At least as a rogue juror. It's even rare in movie logic.
It's still possible to compromise a jury, but only after it's selected, and that is also extremely rare; if you're sure juror 12 is compromised he's remarkable composed for someone either approached for a massive bribe or whose family is being held hostage, which is how such things were actually done the few times it shows up.
1Thanks for your reply, and (re-)welcome to LW!
My conclusion is that I'm pretty sure you're wrong in ways that are fun and useful to discuss!
I hope so! Let's discuss.
(Jsyk you can spoiler possible spoilers on Desktop using ">!" at the start of paragraphs, in case you want to make sure no LWers are spoiled on the contents of a most-of-a-century-old play.)
Regarding the witnesses:
I agree - emphatically! - that eyewitness testimony is a lot less reliable than most people believe. I mostly only brought the witnesses up in my discussion because I thought the jury dismissed them for bad reasons, instead of a general policy of "eyewitnesses are unreliable". (In retrospect, I could have been a lot clearer on this point.)
Regarding the knife:
I agree that the knife being unique would have made things a lot more clear-cut, but disagree about the implications.
If no-one is deliberately trying to frame the accused, the odds of the real killer happening to use the same brand of knife as the one he favors are very low. (What fraction of knives* available to potential suspects are of that exact type? One in a hundred, maybe? If we assume no frame-up or suicide and start with your prior probability of 10% then a naive Bayesian update and a factor of 100 moves that to >90% even without other evidence**.)
If he is actively being framed . . . that's not overwhelmingly implausible, since it's not a secret what kind of knife he uses, and the real killer would be highly motivated to shift blame. However, the idea that he'd have lost his knife, by coincidence, at the same time that someone was using an exact duplicate to frame him (and then couldn't find it afterwards, even though it would be decisive for his defense) . . . strains credulity. I'm less sure about how to quantify the possibility a real killer took his knife without him knowing, got into the victim's apartment, and performed the kill all while the accused was out at the movies; but I feel pretty confident the accused's knife was the murder weapon.
*I'm ignoring the effects of the murder weapon being a knife at all because they're surprisingly weak. The accused owns a knife and favors using it, but so would many alternative suspects; and the accused cohabiting with the victim implies he also has easy access to many alternative methods - poison, arranging an accident - that Hypothetical Killer X wouldn't.
**Full disclosure, I didn't actually perform the calculation until I started writing this post; I admit to being surprised by how little a factor of ~100 changes a ~10% prior probability, though I still feel it's a stronger effect than you're accounting for, and for that matter think your base rates are too low to start with (the fight wasn't just a fight, it was the culmination of years of persistent abuse).
Regarding my conspiracy theories:
I agree that the protagonist having ideological or personal reasons to make the case turn out this way is much more likely than him having been successfully bribed or threatened; aside from anything else, the accused doesn't seem terribly wealthy or well-connected.
I also agree with your analysis of the racist juror's emotional state as presented, though I continue to think it's slightly suspicious that things happened to break that conveniently (the Doylist explanation is of course that the director wanted the bigot to come off as weak and/or needed things to wrap up satisfyingly inside a two-hour runtime, but I'm an incorrigible Watsonian.)
I should probably get into the habit of splitting my comments up. I keep making multiple assertions in a single response, which means when people add (dis)agreement votes I have no idea which part(s) they're (dis)agreeing with.
If model predictions are consistently too low/high:
If using gradient descent, might be a bad starting point
To diagnose: Try running it with number of training rounds and/or learning rate set to/near zero, and seeing if it predicts an unsuitable value for everything.
To fix: Set the starting point to the average outcome in the training set.
If using gradient descent, might be numerical instability
To diagnose: Watch how individual and aggregate predictions change from round to round. If they flicker back and forth (with unornamented gradient descent) or swing back and forth like a pendulum (with momentum), it’s instability.
To fix: Lower learning rate; possibly increase number of rounds to compensate.
Might be distribution shift
To diagnose: See if problem is present in training set too, or just test set; if the latter, you’re looking at distribution shift.
To fix: Extend trends in existing data outwards to the future; add or otherwise adjust-by correction factors; alternately, give up and get more applicable data.
If highs are too high and lows are too low:
This is the default
To diagnose: Confirm whether you are using an MLE-based modelling algorithm (you are), are using a finite amount of data (you are), are modelling a process which isn’t perfectly predictable from its explanatory variables (you are), and exist in reality (you do). If these things are true, then yes, this will happen, and the only question is how hard it’s happening and whether/to what extent you want to correct for it.
To fix: Apply a penalty term.
Might also be change in parameters
To diagnose: Check whether explanatory variables behave very differently in training and test sets.
To fix: Try not making mistakes, and/or fixing them when you do make them.
Is almost guaranteed to happen due to distribution shift
To diagnose: Test out of context (i.e. NOT random-split) and see how far apart the lines on your AvE graphs get.
To fix: Apply a larger penalty term, and/or post-hoc adjustments, until lines align again. Or just get more relevant data.
If highs are too low and lows are too high:
Could, conceivably, be undertraining and undercomplication
To diagnose: See if more training and model complexity improves performance on a true outsample.
To fix: Use more training and model complexity.
Is much more likely to be you evaluating on the training set, or doing something isomorphic
To diagnose: Check whether you’re evaluating on the training set, or doing something isomorphic.
To fix: Don’t evaluate on the training set, or do anything isomorphic.
Is almost certainly NOT happening due to distribution shift
To diagnose: I . . . guess you’d look at the difference between training and deployment? And see whether the inevitable apparent underfit on the training set is actually greater in the test set?
To fix: If you see this happening to a meaningful extent, your modelling context is cursed. Don’t try anything clever, just get a more relevant dataset – or a less cursed project – and don’t look back.
If highs and lows are both making the same error, and middle predictions are making the opposite error:
Could be an inappropriate linkage
To diagnose: Try other linkages and see if they work better. In particular, if you’re using additive(/unity) linkage to predict a price, and see both your highs and lows are too low, try multiplicative(/log) linkage.
To fix: If other linkages work better, use them.
Could be a bound, or other effect of output on output
To diagnose: If the best linkage still isn’t good enough, but the problem is still present in the training set, it’s probably this.
To fix: Just increase model complexity until the problem goes away. Model complexity is a complete and appropriate solution to this problem. You don’t have to do anything else. [Intended affect: hostage reading ransom note on camera.]
If zigzag:
Could be multiple bounds, or multiple other effects of output on output
To diagnose: If you’re sure it’s not noise and it consistently looks like this I have no other interpretation.
To fix: Again, just raise model complexity. [Intended affect: hostage rereading part of ransom note because kidnappers say they didn’t enunciate right the first time.]
I used to implicitly believe that when I have a new idea for a creative(/creative-adjacent) project, all else being equal, I should add it to the end of my to-do list (FIFO). I now explicitly believe the opposite: that the fresher an idea is, the sooner I should get started making it a reality (LIFO). This way:
- I get to use the burst of inspired-by-a-new-idea energy on the project in question.
- I spend more time working on projects conceived by a me with whom I have a lot in common.
The downsides are:
- Some old ideas will end up near the bottom of the pile until I die or the Singularity happens. (But concepts are cheaper than execution, and time is finite.)
- I get less time to polish ideas in my head before committing pen to paper. (But maybe that's good?)
Thoughts on this would be appreciated.
Personally I use a mix of heuristics based on how important the new idea is, how rapid it is and how painful it will be to execute it in the future once the excitement dies down.
The more ADHD you are and the more the "burst of inspired-by-a-new-idea energy" effect is strong, so that should count.
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