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About detecting things going wrong
I was in a relatively-civil discussion on Quora with someone who was deeply worried about individual voter fraud, and it led me to the following reflections.
People frequently encounter the need to detect when something is going wrong, and take some sort of action in response. Consider a primitive cave-dweller listening for predators among all the noises in the night, or a doctor screening for cancer among patients, or a professor looking for cheating students, or a border official looking for criminals and terrorists among people trying to enter the country, or a judge or jury deciding whether the accused is guilty or innocent, or an election official looking for illegitimate or multiple voters.
These situations all have a certain probability of error: you can usually get the probability of error down by spending more time, energy, or money, but getting it down to zero requires infinitely much time, energy, or money. These situations also have in common that there are two kinds of errors: false negatives, in which you fail to spot something that's wrong and take appropriate action, and false positives, in which you flag as wrong something that isn't actually wrong, taking action unnecessarily. (Note that it's impossible to have more false negatives than the actual rate of bad things happening, and impossible to have more false positives than the actual rate of good things happening.) It would of course be nice to eliminate both of these, but as mentioned before that's usually infinitely expensive. Any chucklehead can develop a system with zero false negatives: treat everything as the bad case, and always take action accordingly. And any chucklehead can develop a system with zero false positives: treat nothing as the bad case, and never take the action. Both of those extremes are generally stupid policies; in practice we typically want to balance the risks and costs of false negatives and false positives.
But it's not obvious what "balancing" means. Do we aim for equal accuracy in actual good cases as in actual bad cases? Do we aim for equal confidence in our "good" and "bad" verdicts? Do we aim for equal expected costs of false negatives and false positives? Do we aim to minimize the combined expected costs of false negatives and false positives?
Suppose, for example, the bad thing actually happens 10% of the time, and you assign costs of $1 to a false positive and $10 to a false negative.
The first chucklehead's always-pessimistic strategy produces 10% true positives and 90% false negatives. You are 100% accurate in bad cases, and 0% accurate in good cases. You never produce a "good" verdict, so there's no question of confidence in your "good" verdicts, but only 10% of your "bad" verdicts are correct. Expected cost $.90. This is terrible.
The second chucklehead's always-optimistic strategy produces 10% false negatives and 90% true negatives. You are 100% accurate in good cases, and 0% accurate in bad cases. You never produce a "bad" verdict, so there's no question of confidence in your "bad" verdicts", and 90% of your "good" verdicts are correct. Expected cost $1.00. This is also terrible.
Suppose you come up with an intermediate strategy that correctly classifies 80% of actually-good cases and 80% of actually-bad cases (balancing the accuracies). You'll now get 8% correct positives, 2% false negatives, 72% correct negatives, and 18% false positives. Of all your "bad" verdicts, 8/26 ~= 30% are correct, while of all your "good" verdicts, 72/74 ~= 97% are correct, so you're much more confident in your "good" verdicts than your "bad" verdicts. Expected cost is $.20 from false negatives and $.18 from false positives, nearly balanced, for a total of $.38.
Let's try a somewhat more optimistic strategy that correctly classifies only 70% of actually-bad cases, but 90% of actually-good cases. You'll now get 7% correct positives, 3% false negatives, 81% correct negatives, and 9% false positives. 39% of your "bad" verdicts are correct, 96% of your "good" verdicts are correct, and the expected cost is $.30 from false negatives and $.09 from false positives, for a total of $.39.
How about a somewhat more pessimistic strategy that correctly classifies 90% of actually-bad cases, but 70% of actually-good cases? You'll now get 9% correct positives, 1% false negatives, 63% correct negatives, and 27% false positives. 25% of your "bad" verdicts are correct, 98% of your "good" verdicts are correct, and the expected cost is $.10 from false negatives and $.27 from false positives, for a total of $.37.
Of course, the expected-value calculations change if the costs of false negatives and false positives change. If, for example, we assign $1 cost to both false negatives and false positives, then the always pessimistic strategy still has an expected cost of $.90, but the always optimistic strategy has an expected cost of $.10, the balanced-accuracy strategy has an expected cost of $.20, the somewhat-optimistic strategy has an expected cost of $.12, and the somewhat-pessimistic strategy has an expected cost of $.28. That's right: the mindless always-optimistic strategy actually has a lower expected cost than any of the cleverer strategies in this scenario.
In general (and as you would expect by common sense), the more expensive false negatives are relative to false positives, the more you'll be inclined towards pessimism, and the more expensive false positives are relative to false negatives, the more you'll be inclined towards optimism. Likewise, the more common bad cases are in reality, the more you'll be inclined towards pessimism, while the rarer bad cases are in reality, the more you'll be inclined towards optimism. If the bad case is extremely rare and not terribly expensive, you may be better off ignoring it entirely (and saving the cost of making the decision, which I haven't even mentioned in the above).
Now, how does this play out in the various real-world examples I mentioned?
For the primitive cave-dweller, the cost of a false negative in predator detection is a good chance of being eaten -- a very high price. The cost of a false positive is wasting energy fighting or fleeing things that are actually harmless -- a lower, but nonzero, price. So the cave-dweller will probably, and rationally, select a strategy that produces a lot more false positives than false negatives.
For the cancer-screening doctor, the cost of a false negative is failing to treat a cancer patient, who then suffers or dies unnecessarily. (If you don't really have a treatment for this kind of cancer, that cost is small; likewise if it's a slow-growing cancer detected in an octogenarian who's probably going to die of something else before the cancer gets her.) The cost of a false positive is wasting a bunch of time and money (and suffering -- most cancer treatments are No Fun) treating somebody who didn't need to be treated. Nonetheless, dying of cancer is sufficiently horrible, and we have enough treatments that actually help if applied early, that we still put a high cost on false negatives, while trying to reduce the cost of false positives by not jumping directly to treatment, but first administering a different screening test, perhaps more expensive than the first, to confirm the diagnosis.
For the cheating-detecting professor, the cost of a false negative is students coasting through the class without demonstrating that they've actually learned the material, so that in subsequent classes or jobs they get in trouble and this reflects badly on the professor or the school. All very indirect, and not an enormous cost. The cost of a false positive is going through a nasty dispute-resolution process with the department chair, the dean, the student's parents, etc. (which happens to also be the cost of a true positive!), as well as hurting the reputation of an honest student. So rationally, the professor should probably lean in favor of false negatives over false positives, and might actually be best off not even trying to detect cheating. (Which gives rise to another phenomenon: word gets around, and the actual rate of cheating rises in that professor's courses.)
For the terrorist-detecting border official, the cost of a false negative is a terrorist entering the country and committing an act of mass violence -- quite a high price. The cost of a false positive is the opportunity cost of a potentially valuable tourist or long-term resident, plus the cost to the rejected border-crosser of having to go back where (s)he came from -- torture or death for some, an impediment to the careers of others, and just an annoyance to still others. Of course, if you define "cost" purely in terms of the receiving country, and ignore costs imposed on anyone who isn't already here, you can ignore the cost to the would-be border-crosser. The actual incidence of terrorists trying to cross the border (as tourists, as asylum-seekers, etc.) is probably on the order of one in a million, so depending on how you measure the cost of a false positive, you should be either very optimistic or fairly optimistic (bearing in mind that if you're too optimistic, word will get around and the actual rate of terrorists will increase.)
For the criminal justice system, the cost of a false negative is a bad guy going free and potentially continuing to commit crimes, causing suffering to other members of society, and a loss of public faith in the system. The cost of a false positive is locking up an innocent person, which obviously causes considerable suffering to that person and his/her family, plus the cost of incarceration, plus the opportunity cost of that person being a productive member of society, plus a loss of public faith in the system. In practice, a justice system has several layers that make different trade-offs: the standards for arresting somebody are pessimistic, allowing a lot of false positives; the standards for formally charging someone allow for fewer false positives and more false negatives; and the standards for criminal conviction are "beyond a reasonable doubt", i.e. we want 95+% confidence in our "guilty" verdicts, and are willing to accept a lower degree of confidence in our "innocent" verdicts, i.e. we're willing to accept some false negatives in order to reduce false positives. (At least, that's the way it's supposed to work. In practice, a lot of "guilty" verdicts come from plea deals rather than actual trials, and the system uses the convenient fiction that nobody innocent would ever plead guilty, so we don't need to worry about false positives and therefore get a lot of them.)
For the fraud-detecting voting official, the cost of a false negative is somebody casting a vote who shouldn't be allowed to, or somebody casting an extra vote, in either case changing the vote total by one from what it "should" be and decreasing public faith in the system. The cost of a false positive is ignoring or preventing the vote of somebody who should be allowed to vote, also changing the vote total by one and also decreasing public faith in the system. In other words, the costs of false positives and false negatives are roughly equal. The actual incidence of individual voter fraud, as far as we know, is on the order of one in a million, so a rational expected-cost-minimizing strategy would be extremely optimistic. If it were purely, mindlessly optimistic, with no enforcement whatsoever, that might encourage more people to do it, but as long as the benefit of fraud to the individual voter is minuscule and the cost to them non-trivial (say, waiting in multiple lines), the actual incidence will probably still be pretty low.
In general, political conservatives place a high cost on false negatives and a much lower cost on false positives, which leads them to more-pessimistic strategies -- making it easier to convict accused criminals, exclude foreigners from the country, and potential voters from voting -- while political liberals place a higher cost on false positives, which leads them to the opposite strategies. (There are exceptions on both sides: conservatives in the U.S. appear, to liberals, more concerned about the cost of unnecessarily denying someone a gun than the cost of a dangerous person having a gun, while liberals in the U.S. appear, to conservatives, more concerned about the cost of letting a sexual predator go free than the cost of a false accusation ruining the accused's life.)
In the case of voting, the general tendencies in the previous paragraph happen to align perfectly with political self-interest on both sides: if you make it harder to vote, the first votes you lose tend to be for liberal candidates. From the perspective of political self-interest, false positives don't just have a low cost but actually benefit conservatives, while false negatives might actually benefit liberals (although the number of false negatives is limited by the number of actual fraud attempts, which is probably too few to swing many elections). I'd like to think it objectively obvious that the costs of false positives and false negatives in voter-screening are equal, and the actual incidence of fraud is extremely low (maybe I'm terribly wrong, and it's only one in a thousand rather than one in a million), so any objective, politically unbiased analysis would lead to very little action to prevent individual voter fraud. But it's really hard in today's politically-polarized society for anybody to do an objective, politically unbiased analysis of anything.
People frequently encounter the need to detect when something is going wrong, and take some sort of action in response. Consider a primitive cave-dweller listening for predators among all the noises in the night, or a doctor screening for cancer among patients, or a professor looking for cheating students, or a border official looking for criminals and terrorists among people trying to enter the country, or a judge or jury deciding whether the accused is guilty or innocent, or an election official looking for illegitimate or multiple voters.
These situations all have a certain probability of error: you can usually get the probability of error down by spending more time, energy, or money, but getting it down to zero requires infinitely much time, energy, or money. These situations also have in common that there are two kinds of errors: false negatives, in which you fail to spot something that's wrong and take appropriate action, and false positives, in which you flag as wrong something that isn't actually wrong, taking action unnecessarily. (Note that it's impossible to have more false negatives than the actual rate of bad things happening, and impossible to have more false positives than the actual rate of good things happening.) It would of course be nice to eliminate both of these, but as mentioned before that's usually infinitely expensive. Any chucklehead can develop a system with zero false negatives: treat everything as the bad case, and always take action accordingly. And any chucklehead can develop a system with zero false positives: treat nothing as the bad case, and never take the action. Both of those extremes are generally stupid policies; in practice we typically want to balance the risks and costs of false negatives and false positives.
But it's not obvious what "balancing" means. Do we aim for equal accuracy in actual good cases as in actual bad cases? Do we aim for equal confidence in our "good" and "bad" verdicts? Do we aim for equal expected costs of false negatives and false positives? Do we aim to minimize the combined expected costs of false negatives and false positives?
Suppose, for example, the bad thing actually happens 10% of the time, and you assign costs of $1 to a false positive and $10 to a false negative.
The first chucklehead's always-pessimistic strategy produces 10% true positives and 90% false negatives. You are 100% accurate in bad cases, and 0% accurate in good cases. You never produce a "good" verdict, so there's no question of confidence in your "good" verdicts, but only 10% of your "bad" verdicts are correct. Expected cost $.90. This is terrible.
The second chucklehead's always-optimistic strategy produces 10% false negatives and 90% true negatives. You are 100% accurate in good cases, and 0% accurate in bad cases. You never produce a "bad" verdict, so there's no question of confidence in your "bad" verdicts", and 90% of your "good" verdicts are correct. Expected cost $1.00. This is also terrible.
Suppose you come up with an intermediate strategy that correctly classifies 80% of actually-good cases and 80% of actually-bad cases (balancing the accuracies). You'll now get 8% correct positives, 2% false negatives, 72% correct negatives, and 18% false positives. Of all your "bad" verdicts, 8/26 ~= 30% are correct, while of all your "good" verdicts, 72/74 ~= 97% are correct, so you're much more confident in your "good" verdicts than your "bad" verdicts. Expected cost is $.20 from false negatives and $.18 from false positives, nearly balanced, for a total of $.38.
Let's try a somewhat more optimistic strategy that correctly classifies only 70% of actually-bad cases, but 90% of actually-good cases. You'll now get 7% correct positives, 3% false negatives, 81% correct negatives, and 9% false positives. 39% of your "bad" verdicts are correct, 96% of your "good" verdicts are correct, and the expected cost is $.30 from false negatives and $.09 from false positives, for a total of $.39.
How about a somewhat more pessimistic strategy that correctly classifies 90% of actually-bad cases, but 70% of actually-good cases? You'll now get 9% correct positives, 1% false negatives, 63% correct negatives, and 27% false positives. 25% of your "bad" verdicts are correct, 98% of your "good" verdicts are correct, and the expected cost is $.10 from false negatives and $.27 from false positives, for a total of $.37.
Of course, the expected-value calculations change if the costs of false negatives and false positives change. If, for example, we assign $1 cost to both false negatives and false positives, then the always pessimistic strategy still has an expected cost of $.90, but the always optimistic strategy has an expected cost of $.10, the balanced-accuracy strategy has an expected cost of $.20, the somewhat-optimistic strategy has an expected cost of $.12, and the somewhat-pessimistic strategy has an expected cost of $.28. That's right: the mindless always-optimistic strategy actually has a lower expected cost than any of the cleverer strategies in this scenario.
In general (and as you would expect by common sense), the more expensive false negatives are relative to false positives, the more you'll be inclined towards pessimism, and the more expensive false positives are relative to false negatives, the more you'll be inclined towards optimism. Likewise, the more common bad cases are in reality, the more you'll be inclined towards pessimism, while the rarer bad cases are in reality, the more you'll be inclined towards optimism. If the bad case is extremely rare and not terribly expensive, you may be better off ignoring it entirely (and saving the cost of making the decision, which I haven't even mentioned in the above).
Now, how does this play out in the various real-world examples I mentioned?
For the primitive cave-dweller, the cost of a false negative in predator detection is a good chance of being eaten -- a very high price. The cost of a false positive is wasting energy fighting or fleeing things that are actually harmless -- a lower, but nonzero, price. So the cave-dweller will probably, and rationally, select a strategy that produces a lot more false positives than false negatives.
For the cancer-screening doctor, the cost of a false negative is failing to treat a cancer patient, who then suffers or dies unnecessarily. (If you don't really have a treatment for this kind of cancer, that cost is small; likewise if it's a slow-growing cancer detected in an octogenarian who's probably going to die of something else before the cancer gets her.) The cost of a false positive is wasting a bunch of time and money (and suffering -- most cancer treatments are No Fun) treating somebody who didn't need to be treated. Nonetheless, dying of cancer is sufficiently horrible, and we have enough treatments that actually help if applied early, that we still put a high cost on false negatives, while trying to reduce the cost of false positives by not jumping directly to treatment, but first administering a different screening test, perhaps more expensive than the first, to confirm the diagnosis.
For the cheating-detecting professor, the cost of a false negative is students coasting through the class without demonstrating that they've actually learned the material, so that in subsequent classes or jobs they get in trouble and this reflects badly on the professor or the school. All very indirect, and not an enormous cost. The cost of a false positive is going through a nasty dispute-resolution process with the department chair, the dean, the student's parents, etc. (which happens to also be the cost of a true positive!), as well as hurting the reputation of an honest student. So rationally, the professor should probably lean in favor of false negatives over false positives, and might actually be best off not even trying to detect cheating. (Which gives rise to another phenomenon: word gets around, and the actual rate of cheating rises in that professor's courses.)
For the terrorist-detecting border official, the cost of a false negative is a terrorist entering the country and committing an act of mass violence -- quite a high price. The cost of a false positive is the opportunity cost of a potentially valuable tourist or long-term resident, plus the cost to the rejected border-crosser of having to go back where (s)he came from -- torture or death for some, an impediment to the careers of others, and just an annoyance to still others. Of course, if you define "cost" purely in terms of the receiving country, and ignore costs imposed on anyone who isn't already here, you can ignore the cost to the would-be border-crosser. The actual incidence of terrorists trying to cross the border (as tourists, as asylum-seekers, etc.) is probably on the order of one in a million, so depending on how you measure the cost of a false positive, you should be either very optimistic or fairly optimistic (bearing in mind that if you're too optimistic, word will get around and the actual rate of terrorists will increase.)
For the criminal justice system, the cost of a false negative is a bad guy going free and potentially continuing to commit crimes, causing suffering to other members of society, and a loss of public faith in the system. The cost of a false positive is locking up an innocent person, which obviously causes considerable suffering to that person and his/her family, plus the cost of incarceration, plus the opportunity cost of that person being a productive member of society, plus a loss of public faith in the system. In practice, a justice system has several layers that make different trade-offs: the standards for arresting somebody are pessimistic, allowing a lot of false positives; the standards for formally charging someone allow for fewer false positives and more false negatives; and the standards for criminal conviction are "beyond a reasonable doubt", i.e. we want 95+% confidence in our "guilty" verdicts, and are willing to accept a lower degree of confidence in our "innocent" verdicts, i.e. we're willing to accept some false negatives in order to reduce false positives. (At least, that's the way it's supposed to work. In practice, a lot of "guilty" verdicts come from plea deals rather than actual trials, and the system uses the convenient fiction that nobody innocent would ever plead guilty, so we don't need to worry about false positives and therefore get a lot of them.)
For the fraud-detecting voting official, the cost of a false negative is somebody casting a vote who shouldn't be allowed to, or somebody casting an extra vote, in either case changing the vote total by one from what it "should" be and decreasing public faith in the system. The cost of a false positive is ignoring or preventing the vote of somebody who should be allowed to vote, also changing the vote total by one and also decreasing public faith in the system. In other words, the costs of false positives and false negatives are roughly equal. The actual incidence of individual voter fraud, as far as we know, is on the order of one in a million, so a rational expected-cost-minimizing strategy would be extremely optimistic. If it were purely, mindlessly optimistic, with no enforcement whatsoever, that might encourage more people to do it, but as long as the benefit of fraud to the individual voter is minuscule and the cost to them non-trivial (say, waiting in multiple lines), the actual incidence will probably still be pretty low.
In general, political conservatives place a high cost on false negatives and a much lower cost on false positives, which leads them to more-pessimistic strategies -- making it easier to convict accused criminals, exclude foreigners from the country, and potential voters from voting -- while political liberals place a higher cost on false positives, which leads them to the opposite strategies. (There are exceptions on both sides: conservatives in the U.S. appear, to liberals, more concerned about the cost of unnecessarily denying someone a gun than the cost of a dangerous person having a gun, while liberals in the U.S. appear, to conservatives, more concerned about the cost of letting a sexual predator go free than the cost of a false accusation ruining the accused's life.)
In the case of voting, the general tendencies in the previous paragraph happen to align perfectly with political self-interest on both sides: if you make it harder to vote, the first votes you lose tend to be for liberal candidates. From the perspective of political self-interest, false positives don't just have a low cost but actually benefit conservatives, while false negatives might actually benefit liberals (although the number of false negatives is limited by the number of actual fraud attempts, which is probably too few to swing many elections). I'd like to think it objectively obvious that the costs of false positives and false negatives in voter-screening are equal, and the actual incidence of fraud is extremely low (maybe I'm terribly wrong, and it's only one in a thousand rather than one in a million), so any objective, politically unbiased analysis would lead to very little action to prevent individual voter fraud. But it's really hard in today's politically-polarized society for anybody to do an objective, politically unbiased analysis of anything.
no subject
Yes, the Quora discussion that instigated this was largely about voter suppression vs. fraud prevention. I pointed out a bunch of specific government actions -- shutting down polling places in predominantly-black neighborhoods but not predominantly-white neighborhoods, shutting down transit-accessible polling places but not car-accessible polling places, shutting down early in-person voting that's used largely by blacks but not mail-in voting that's used largely by whites, etc. -- that appeared to me to be racial and/or partisan voter suppression, and the other participant in the discussion felt I was just assuming voter suppression without any actual evidence.
At some point in that discussion I said something like "Given the choice between more people voting and fewer people voting, I'll take the former because democracy works better when we hear from more people. Too many people voting is the least of America's problems." My debate partner took this as an endorsement of voter fraud and illegal voting, so I clarified: "As far as I know, neither major party supports illegal voting. But by and large, Republicans want fewer people to be legally allowed to vote, and fewer of those people to actually vote, while Democrats, by and large, want more people to be legally allowed to vote, and more of those to actually vote. I think the latter is objectively more compatible with the idea of democracy."
no subject
All of which doesn't tell us anything about whether we need to crack down on voter ID at the polls, because that wouldn't have had any effect on this scheme. It does suggest we need to improve security for absentee and mail-in ballots, which (as mentioned above) are used disproportionately by white people.