After the death of Laquan McDonald (which for what it’s worth strikes me as being murder pure and simple, but this has no bearing on what I’m going to be talking about), the Mayor of Chicago created a task force, with the job of scrutinizing the practices of the city’s police department. The task force released its report a few months ago and, brace yourself, it concluded that Chicago Police Department was plagued with racism. The media, both local and national, from the Chicago Tribune to the New York Times, reported that conclusion uncritically. But I have read the relevant section of the report and, of course, it doesn’t demonstrate anything of the sort. As every report of that sort, it has absolutely no scientific worth, but is just a political document whose only purpose is to placate the populace and prevent riots.I want to focus on a particular fallacy that the authors of that report committed, because it’s one of the most common fallacies that people use to conclude that racism is rampant in law enforcement. (For instance, it can also be found in the Department of Justice’s report on Ferguson Police Department, which was released after the death of Michael Brown.) Moreover, I think it’s also interesting from a purely logical/statistical point of view, if you’re into that kind of things. Finally, it illustrates very nicely how the lack of political diversity in social science can result in bad consequences, such as systematic error.
The argument is nicely summarized in this article, which was published in the New York Times in 2001, showing that it’s a fallacy with a long history:
It is no longer news that racial profiling occurs; study after study over the past five years has confirmed that police disproportionately stop and search minorities. What is news, but has received virtually no attention, is that the studies also show that even on its own terms, racial profiling doesn’t work.
Those who defend the police argue that racial and ethnic disparities reflect not discrimination but higher rates of offenses among minorities. Nationwide, blacks are 13 times more likely to be sent to state prisons for drug convictions than are whites, so it would seem rational for police to assume that all other things being equal, a black driver is more likely than a white driver to be carrying drugs.But the racial profiling studies uniformly show that this widely shared assumption is false. Police stops yield no significant difference in so-called hit rates — percentages of searches that find evidence of lawbreaking — for minorities and whites. If blacks are carrying drugs more often than whites, police should find drugs on the blacks they stop more often than on the whites they stop. But they don’t.
So, to be clear, the argument roughly goes as follows:
(1) When you look at the hit rate (i. e. the proportion of people stopped who carried contraband) for whites, you find that it’s higher than for blacks. (The article I just quoted only says that it was not lower, which is a weaker claim, but the stronger claim is made in the reports I mentioned above.)
(2) Therefore, the offending rate for blacks is lower than for whites or, at least, is not higher.
(3) Yet black people are disproportionately stopped by the police.
(4) So, by inference to the best explanation, the police is racist.
I know that sounds pretty convincing, but the inference from (1) to (2) is actually a statistical fallacy, as I’m now going to explain.
The problem is that, for (2) to follow from (1), a necessary condition is that people are stopped randomly by the police or, more generally, that the sampling methods used by cops to stop blacks and whites, together with facts about the structure of each group, make 
an unbiased estimator of 
where 
is the hit rate for whites, 
the hit rate for blacks, 
the offending rate for whites and 
the offending rate for blacks. Let’s call that hidden premise (1*), so I can refer to it quickly later. What (1*) means, roughly, is that if you looked at 
samples of people stopped by the police and recorded the hit rate for blacks and whites in each case, the mean of 
over those samples would tend to 
as increases.
an unbiased estimator of where is the hit rate for whites, the hit rate for blacks, the offending rate for whites and the offending rate for blacks. Let’s call that hidden premise (1*), so I can refer to it quickly later. What (1*) means, roughly, is that if you looked at samples of people stopped by the police and recorded the hit rate for blacks and whites in each case, the mean of over those samples would tend to as increases.So, in order to be able to infer (2), it is not enough that (1) be true, it must also be the case that (1*) is true. The problem is that, not only do we have no reason to believe that (1*) is true, but on the contrary there is absolutely no doubt that it’s false. Indeed, nobody questions that cops are biased against blacks, in the neutral sense that they stop blacks disproportionately. That’s basically what (3) says in the argument, which everyone agrees is true. The question is whether that bias results from a true belief that the offending rate is higher among blacks than among whites or, as (4) asserts, from prejudice on the part of cops. (I’m using “racist” to mean something like “discriminate against black people because they are prejudiced”. This may not be a good definition of “racism”, but that’s irrelevant to the point I’m trying to make, for the argument I’m attacking clearly purports to show that cops stop black people more often not because they actually commit more crimes but rather because they are prejudiced against them.) But, whether or not (4) is true, as long as (3) is true, (1*) is probably false. Indeed, as long as (3) is true, it’s likely that overestimates , quite possibly to the point that comes out as positive even though is actually negative.
overestimates , quite possibly to the point that comes out as positive even though is actually negative.Just think about it for a moment: if cops believe — rightly or wrongly, it doesn’t matter here — that blacks are significantly more likely than whites to carry contraband or engage in illegal activities, they are only going to stop whites that have “I’m a criminal” written on their face, whereas in the case of blacks they’re going to cast a much wider net. What this means is that the sample of blacks is going to be much more random than the sample of whites, which is going to contain a lot more guilty people than in the white population at large. So, as long as (3) is true, it’s entirely unsurprising that the hit rate for whites is higher than the hit rate for blacks, even if the offending rate is actually much higher for blacks than for whites. Which is why (4) doesn’t follow from (1) and (3).
It may be helpful to consider a silly example to illustrate the problem. Suppose that, in some weird country, cops are stopping people in the street to see if they have coins in their pockets, because in that country it’s illegal to own coins. The cops stop people who have a beard more often than people who don’t because, for some reason, they think people who have a beard are more likely to have coins in their pockets than people who don’t have a beard. Suppose that, in fact, the cops are right, for 10% of people who have a beard have coins in their pockets while only 5% of people who don’t have a beard do. Suppose, moreover, that among people who don’t have a beard, they only stop people who have a red shirt, while they stop people who have a beard randomly. Perhaps it’s because, for whatever reason, they are convinced that, among people who don’t have a beard, people with a red shirt are significantly more likely to have coins in their pockets. As it turns out, the cops are right again, for although only 5% of people who don’t have a beard have coins in their pockets, 15% of those who don’t have a beard and wear a red shirt do.
A sociologist observes them and notice that, not only do they stop people with a beard more often, but among the people they stop, those who have a beard are less likely to have coins in their pockets than people who don’t have a beard. Indeed, among the people they stop, only 10% of the people who have a beard have coins in their pockets, while 15% of those who don’t have a beard do. He infers that people who have a beard are less likely to have coins in their pocket and, since the cops nevertheless stop them more often than people who don’t have a beard, he concludes that they are prejudiced against people who have a beard and that’s why they stop them more often than people who don’t. Except that, as we have seen, he is wrong. People who have a beard are more likely to have coins in their pockets and, if cops stop them more often, it’s because they know it. The sociologist just committed the fallacy I explained above.
Now just pause to think about what this means exactly. It’s not just that the argument is fallacious, it’s fallacious in about the worst way an argument can be fallacious. Indeed, (4) only follows from (3) if (2) is true, but (2) only follows from (1) if (1*) is true. However, as we have just seen, if (3) is true then (1*) is probably not true. Thus, 2 of the argument’s premises — (1*) and (3) — are incompatible, which makes it a particularly defective argument, to say the least… Moreover, since I’m using “racist” in (4) to mean something like “have bias and that bias results from prejudice”, (4) entails (3). So the problem with the argument is not just that it contains a hidden premise that happens to be false, it’s that if the conclusion of the argument is true, then the argument is not valid and cannot possibly establish the conclusion! And that’s what people who use that argument have not realized: even if cops were prejudiced in that sense, you couldn’t show it by using that argument, for the reason I just pointed out.
Before I stop writing, I want to make two remarks. First, how come that argument is so common even on the part of professional social scientists, given that it’s guilty of a relatively basic statistical fallacy? Of the two authors of the NYT article, at least one of them — John Lamberth, who is a psychologist — surely knows enough statistics to understand the problem with that argument, yet he committed that fallacy all the same and, since I don’t think he is dishonest, I think it’s very likely that he really didn’t notice that his argument was fallacious. Moreover, I’m sure that I’m not the first person to notice the problem with that argument, yet I have never seen anyone point it out. Anyone who has taken an introductory class on statistics knows enough to detect that fallacy, but even people who should know better don’t notice it or, if they do, they don’t say anything.
A pretty natural explanation is that liberals are biased in favor of the hypothesis that cops are racist and the vast majority of social scientists are liberal, so either they are blind to the problem with their argument or, if they notice it (as I’m sure that some have), they don’t say anything because they’re afraid that they would be punished for it. And the truth is that they are right to be afraid, because they would be punished. The fact that this terrible argument is used over and over, even by people who should clearly know better, is a very good illustration of the damages that ideological uniformity can do. If there were more conservatives in the social sciences, enough people would have pointed out the fallacious character of that argument for people to stop using it, because they would have noticed the problem and they wouldn’t have been afraid to talk about it.
Of course, this goes beyond the scientific community, it’s a widespread problem in the culture at large. Race is such a taboo in the US, even more so than in Europe, that it’s absolutely impossible to have a honest conversation about any aspect of the racial problem in this country. And liberals are largely to blame for that state of affairs. You sure as hell aren’t going to solve any problem by burying your head in the sand and vilifying anyone who says inconvenient truths, even when by your own admission you would be incapable of showing that they are wrong. Liberals are prone to talk about the dangers of having a honest conversation about race. But they also completely ignore the dangers of not having a honest conversation about race. I have never seen a convincing argument that the former outweigh the latter. But I guess that’s a topic for another post…
The last remark I wanted to make is that, of course, even if blacks are in fact more likely to commit crimes (which they most certainly are, but that’s also a topic for yet another post), it doesn’t follow that racial profiling is morally justified. My own view of the question is that it’s a very complicated issue, much more than liberals generally assume (because they never consider the costs of abandoning every kind of racial profiling), but also one which I think is kind of pointless to discuss. Indeed, the fact of the matter is that, as long as blacks will commit significantly more crimes on average than whites, the police will engage in some kind of statistical discrimination against them. So, if you want to insist that racial profiling is unambiguously wrong, I’m happy to grant you that because it has little to no bearing on any practical issue. I think a much more interesting question is: how do we make sure that blacks no longer commit more crime on average than whites? I think progressives have potentially a lot of things to say about that and, to be fair, some of them do. But, in order to ask that question and find a solution, you must first be able to have a honest discussion about the facts, which isn’t going to happen as as long as liberals keep treating dissent like the Spanish Inquisition.
EDIT: I wrote a follow-up on this post in which I clarify a few points, so you may want to read it before you comment.
ANOTHER EDIT: It’s interesting that John Lamberth, one of the authors of the article I quote, claimed back in the 1990’s that the state police in New Jersey was racist because it was stopping black drivers for speeding at a disproportionate rate. However, when a more rigorous study was conducted a few years later, it showed that black drivers were more likely to speed and that it explained why they were pulled over more often. The study in question was actually commissioned by the state’s Justice Department, which hoped it would bolster its case against the state police. When it turned out that the opposite was true, a lawyer with the Justice Department’s special litigation section tried to block the release of the report, which is another proof that white supremacy is alive and well in the US…
What about neighborhood profiling?
Say every police officer stops N people/day regardless of their race. Assume further that the police department allocates officers to neighborhoods based on the actual crime-rate in that neighborhood.
This has nearly the same effect as racial profiling because neighborhoods are often segregated by race (for a variety of usually non-nefarious reasons) and race is correlated with crime. The situation gets “worse” once one accounts for the officer’s behavior: if they know they’ve been assigned to a more crime-prone area they’re probably going to be more proactive/vigilant and make more stops. Or at least I personally hope so.
IIRC this is pretty common in quite a few major cities.
I think anybody who claims that racial profiling is “immoral” needs to explain how to distinguish the case above or describe exactly at which point the above goes from being basic common sense to “racist & evil.”
Well, it seems likely to me that direct racial profiling is less efficient than neighborhood profiling although, as far as the racial disparity in the probability of being stopped is concerned, they may be roughly equivalent. So, if I’m right about that, someone could use that fact to argue that direct racial profiling is not okay but neighborhood profiling is. Liberals often condemn even neighborhood profiling, on the ground that it’s a kind of racial profiling, which in a way it is. I think it’s that kind of things they have in mind when they talk about institutional racism in law enforcement.
To be clear, I don’t think it’s obvious that either kind of racial profiling is morally permissible, let alone obligatory. But I definitely agree with you that it’s also not obvious that it’s not morally permissible or even obligatory. In particular, neighborhood profiling is exactly the kind of things I had in mind when I wrote that it wasn’t clear to me that every kind of racial profiling was unacceptable, because it should be clear to anyone who thinks about it that whether that kind of racial profiling is morally permissible is a non-trivial question.
“Liberals are prone to talk about the dangers of having a honest conversation about race. But they also completely ignore the dangers of not having a honest conversation about race. I have never seen a convincing argument that the former outweigh the latter.”
Conservatives never wait for the approval of liberals before doing other things that they want to do. So why do so in this area? if they want to have what they consider to be honest conversations about race, then they can surely do that– regardless of what liberals think of it. If liberals call conservatives racist, and they are not being so, then they can stand up and say that– or yell it, as the Right Wing TV and video pundits do.
I think it’s a bit naive to say that. Nowadays, the accusation of racism carries such a stigma that, if people can’t say certain things without being accused of racism, it inevitably stifles debate. Even on conservative networks, there are things people won’t say about race, because it’s dangerous. This is particularly true in some environments, such as academia, where I can guarantee you that saying the kind of things I say puts you in a very tight spot. I also think it’s problematic if conservatives can only talk about that kind of things between themselves. This should be the kind of things people can discuss rationally, especially in academia, where people are supposed to study the facts without having to fear they will face repercussions for saying things they have good reasons to believe to be true.
I don’t think point 2 follows from point 1 of your argument.
I wouldn’t expect, “When you look at the hit rate (i. e. the proportion of people stopped who carried contraband) for whites, you find that it’s higher than for blacks.” to imply that “Therefore, the offending rate for blacks is lower than for whites or, at least, is not higher”.
What it implies is that the police force is being inefficient with its stops (and I would call that racism). If pulling over a white person yields a higher chance of finding contraband they should pull over more whites and/or less black people until the hit rate is the same. This is completely consistent even if the offending rate for blacks is higher. In that case they should pull over more blacks but not past the point where the marginal black person pulled over yields a lower offending rate than the marginal white person pulled over.
Well, this is not my argument and I agree with you that 2 doesn’t follow from 1, since it’s precisely the inference I’m rejecting in my post 🙂 However, despite what you say, it doesn’t even follow from 1 that cops are being inefficient, although depending on the details of the case it could arguably make it likely.
Your argument implicitly assumes that, if the hit rate for whites is higher than the hit rate for blacks, then the offending rate for the whites that were not stopped is higher than the hit rate for blacks would be if cops chose to arrest less blacks and more whites. But this is hardly obvious, because it depends on a lot of factors, so I don’t think we can just assume that.
Indeed, suppose that, as I was suggesting in my post, cops only stop a white person when he is a known criminal, whereas they not only stop every black person who is a known criminal but also randomly stop 100 blacks who aren’t known criminals. Suppose, moreover, that among blacks who aren’t know criminals, the offending rate is 10%, while it’s only 5% among whites who aren’t known criminals.
In that case, if cops reduce the number of blacks while increasing the number of whites they stop by only stopping blacks who are known criminals and, in addition to stopping every white who is a known criminal, randomly stopping 100 whites who aren’t known criminals, they will catch less people who have something illegal on them.
Yet this scenario is consistent with the assumption that, given how I have assumed that cops decide to stop people, the hit rate for whites was initially higher than for blacks. It’s even possible that, by changing their procedure to decide who to stop in the manner I have just described, cops will make the hit rate for blacks equal to the hit rate for whites. You just have to make the right assumptions about the number of known criminals that drive in front of them, the race-specific offending rates among them and the racial composition of that group.
Of course, even in that case, you can imagine cops stopping less blacks and more whites in a way that increases the number of people caught with something illegal on them. (Trivially, if cops had stopped every black person they actually stopped who had something illegal on him but none of the black people they stopped who did not, it would have been the case.) But it’s not obvious that cops could actually do that, because the necessary conditions for this to be possible in practice may not be satisfied.
Again, to be clear, I’m not denying that if the hit rates for whites is higher than for blacks even though cops stop more blacks than whites, it could be that cops are being inefficient. (I’m sure there are even cases in which, from the fact that the hit rates for whites is higher than for blacks even though cops stop more blacks than whites, it’s reasonable to infer that cops probably are inefficient.) But whether that is the case depends on a lot of factors, such as the characteristics of blacks and whites in the area, how the observable characteristics the police can use to decide who to stop correlate with criminal behavior in each racial group, etc.
In a way, that cops are being inefficient is trivial, because nobody can reasonably believe that cops are perfect maximizers. The interesting question is whether they are inefficient in a way that make them racist, but I don’t think we can infer this from the fact that the hit rate for whites is higher than for blacks, even if cops stop more blacks than whites. (Again, there are probably cases in which it’s possible to make that inference, but we need more information before we can make it.)
I don’t disagree with your explanation. However, I find it a bit suspect that to demonstrate you use a step function where a white person is either a known criminal or they’re not, and that cops can somehow tell which group that person falls into before pulling them over. I assume cops judge the likelihood that someone is a criminal along a smooth continuous function between 0 and 100% and pull over anyone they judge above x percent. In that case, the fact that whites have a higher hit rate when pulled over means the cops are judging incorrectly, which I call racism. Granted it may be a poor assumption that a cop’s perceived likelihood of being a criminal is smooth and continuous. If it is instead a step function as you describe I agree.
Of course, the model I used to illustrate my point is unrealistic, but I think yours is even less realistic. However, I don’t want to press that point and spend time arguing about it, because the point I was really trying to make is that we just can’t make the inference you were suggesting without making a lot of non-obvious assumptions, since whether that inference is reliable will depend on a lot of things beside the race-specific hit rates and the number of people of each race that are being stopped. And, while I’m sure we can disagree about what assumptions we should make exactly, that much is definitely true.
To tell you the truth, when I started writing this post, I had the same intuition as you and intended to conclude my post by making the point you made in your comment. But then I thought about it for a minute and realized that I just couldn’t make that inference because it rests on a lot of non-obvious assumptions. I suspect that, if we have that intuition, it’s because of a familiarity with maximization problems in economics. But I don’t think we can assume that, in this case, the function we are dealing with has the same properties as the kind of functions economists deal with.
Racial profiling often presents a moral dilemma.
The underlying problem is that race can, in many epistemic contexts, provide key information as to the level of risk of some outcome. Jesse Jackson, rather famously, made this remark:
“There is nothing more painful to me at this stage in my life than to walk down the street and hear footsteps and start thinking about robbery. Then look around and see somebody white and feel relieved…”
This is pretty good example of how, given the exact informational context Jackson describes, the predictive value of race is critical as to what one may do (e.g., continue down the street, vs. try to find a store one can duck into).
Obviously, if one had perfect information about the person walking behind, race would be irrelevant — but one virtually never has perfect information in making choices.
I don’t think any reasonable person would hold Jackson, or anyone else, at fault if they chose to take evasive behavior if the person behind was a young black male, and not do so if it were a young white male. Yet this certainly would seem to fit any definition of racial profiling.
When it comes to officers of the law, being agents of the government, we have higher expectations as to how and when they are allowed to use race in their decisions. Agents of the government are generally expected to ignore race, in the interests of fairness, even if race provides strong evidence of some outcome.
The trick is to find some balance between the unfairness of treating an individual on the basis of race and the risk of not preventing a negative outcome because one has ignored race.
Appreciate the feedback. Perhaps I am overthinking it from an economic perspective. It’s an interesting thought experiment regardless.
Enjoy your blog, love seeing alternative viewpoints that don’t get covered elsewhere!
Thanks for the nice words about my blog and also for your interesting comment. It forced me to explain something which I think is important. After I realized my initial intuition was misguided, I actually considered explaining in my post what I said in my reply to you, but I was too lazy to do it :-p
Because, statistically, blacks offend at a much higher rate than whites, it gives rise to any number of seemingly plausible, but quite fallacious, arguments that racial discrimination must be operating in various criminal justice contexts.
Fairly recently it was claimed that an algorithm designed to predict recidivism, and which did not use race as a predictive variable, must nonetheless be discriminatory, because more blacks than whites were “labeled higher risk, but didn’t reoffend.” But that claim clearly hung on a fallacy, as the following article makes clear:
http://www.theamericanconservative.com/articles/does-pre-crime-have-a-race-problem/
Thanks for the link, I’ll read this later. I suspect it has to do with the disparate impact doctrine, which indeed often leads to absurd results.
You keep saying that is a fact that search hit rates are higher for Whites than Blacks, but that isn’t what even the politicized report you quoted said, which was: “Police stops yield no significant difference in so-called hit rates — percentages of searches that find evidence of lawbreaking — for minorities and whites.”
You show that even if the White hit rate were higher, that still wouldn’t necessarily mean Blacks were discriminated against, which goes beyond a “steel-man” argument, (the opposite of a straw-man, arguing against the best form of your opponent’s thesis) to one that shows that even assuming a falsehood that would be most supportive of your opponents’ thesis, it still fails. Given that the hit rates are indistinguishable despite a higher percentage of Blacks being searched shows directly, without any more complicated argument needed, that Blacks are not being discriminated against in being selected for searches and that they are more likely to carry contraband.
The quote is from a NYT article, where it’s true that the authors make a weaker claim, but you can find the stronger claim in the reports I mentioned before that. I added a note in the post to make that clear, thanks for pointing that out.)
“But the racial profiling studies uniformly show that this widely shared assumption is false. Police stops yield no significant difference in so-called hit rates — percentages of searches that find evidence of lawbreaking — for minorities and whites. If blacks are carrying drugs more often than whites, police should find drugs on the blacks they stop more often than on the whites they stop. But they don’t.
So, to be clear, the argument roughly goes as follows:
(1) When you look at the hit rate (i. e. the proportion of people stopped who carried contraband) for whites, you find that it’s higher than for blacks.”
Your point (1) directly contradicts the claim you quote.
The easiest way to demonstrate that the claim is nonsensical is to note that if the black population were oversampled in the way claimed (more blacks were being stopped on less evidence, and assuming the evidence accurately predicts the hit rate) then the hit rate for blacks would be lower than for whites.
If the hit rates for blacks were actually lower than that for whites… well the following is simply nonsense: “if cops believe — rightly or wrongly, it doesn’t matter here — that blacks are significantly more likely than whites to carry contraband or engage in illegal activities, they are only going to stop whites that have “I’m a criminal” written on their face, whereas in the case of blacks they’re going to cast a much wider net.” The “nonsense” part is the claim that it doesn’t matter whether the cops are wrong. If they are wrong then the action of casting a wider net among blacks is racist and cannot be justified.
I’m guessing that you are alluding to the same point that EH made in his comment above. As I explained to him, it’s true that the authors of the article I quote made a weaker claim, but the reports I mentioned before that make the stronger claim.
Sure, but I wanted to show that, even if the hit rate for white is higher (which it sometimes is), the conclusion still doesn’t follow.
Of course, if the cops are wrong then what they’re doing can’t be justified, but that’s not what I meant when I said that “it doesn’t matter here”. What I meant is that, even if the cops are wrong (and therefore are racist), the inference is still fallacious, because what matters from a purely statistical/logical point of view is that cops don’t sample blacks and whites in the same way, not why they don’t. Obviously, from a moral point of view, it does matter.
Yes, I was making the same point as EH. There was no clue in your post that the two reports mentioned claimed that the hit rates for whites was higher than the hit rates for blacks, so your “to be clear” statement of the proposition seemed to come from left field. Your in-line clarification does fix this.
If the cops are wrong about the contraband rate then their action in oversampling blacks (relative to evidence other than race) is indeed racist, which is the claimed conclusion from a white hit rate higher than black, so the conclusion that cops are racist can be wrong only if blacks are in fact more likely to be carrying contraband.
So the only interesting case is:
(a) black contraband rate higher than white
(b)white hit rate higher than black
…and I’m not following your claim that this is not evidence of racism, absent special circumstances such as the sampling taking place in neighborhoods where the only whites present have contraband rates higher than the local black population (so that we are not really dealing with the interesting case, though it might be falsely claimed that we were).
Now, cops can’t actually legally stop people merely because they believe they are criminals. And if non-contraband-carrying blacks provide more excuses for stopping (e.g., speeding) than non-contraband-carrying whites that might account for more false positives (non-hits) among blacks than whites, but I’d like to see some argument about the magnitude of that effect, or others that might be suggested before dismissing the racism hypothesis. Of course, if the hit rate ends up nonetheless being equal then such facts may actually point to a Ferguson Effect, i.e. cops being more reluctant to stop blacks than they should be.
Fair enough.
Again, I completely agree with that, but nothing I say contradicts it. My point is only that, even if cops are racist because they falsely believe that blacks are more likely to carry contraband than whites, the inference from 1 to 2 is fallacious because the very fact that they are racist and oversample blacks makes the estimator biased.
I think you are making the same point as Kevin above. If you read my exchange with him, I explain why, even when a and b are true, whether the cops are racist depends on the assumptions you make about 1) the nature of the relationship between evidence and the probability that someone is carrying contraband in each group and 2) how evidence is distributed among different groups. I don’t have time to figure out what assumptions are necessary and sufficient for the cops to be racist (i. e. for the way in which they sample people in different groups to result in a suboptimal overall hit rate) when a and b are true, though I think it’s an interesting question, but as I argue in my discussion with Kevin I don’t think we can just assume that sufficient conditions are fulfilled in any realistic case. (I implicitly defined “racism” so that cops are racist if the way in which they sample people in different groups results in a suboptimal overall hit rate, but this definition should probably be strengthened, since it makes it extremely easy for cops to count as racist. Perhaps we should say that the overall hit rate must be suboptimal but not by just a little or something like that.)
To be clear, the purpose of this post isn’t to dismiss the racism hypothesis, it’s only to point out that a common inference whose conclusion is that cops are racists is fallacious. I’m sure cops are often racist to some extent, though as I have argued in other posts, I don’t think racism in the criminal justice systems explains a large part of the racial disparities observed for various outcomes. As you can see in the links I added to my post, there is evidence that blacks are more likely to speed and the effect seems to be large enough to explain the disparity in stopping rates, but I have no idea if this is true in general since I have never looked at the literature on that. (I added this before you wrote your comment, just because it involves John Lamberth and I remembered that I meant to add this to my post, but it just happens to be relevant to what you say in your comment.) I agree that whether or not the police are racist or, on the contrary, are more reluctant to stop blacks than they should be is going to depend on that sort of questions, among other things. There is also no reason to expect that the answer will be the same in different locations across the US. Not only because police departments are different, but so are the public they deal with.
This whole discussion with you and the one I had with Kevin previously makes me think that perhaps I should write another post to sort out these issues. At the very least, I’d like to get clear on how I should define racism for the purposes of this discussion and what conditions are necessary and sufficient to guarantee that cops are racist in that sense when a and b are true, which I don’t have time to think about right now because I’m very busy. But I’ll definite keep that in mind for when I have more time.
YOu should check out papers by KNowles, Persico and Todd, in the Journal of Political Economy (http://nicolapersico.com/files/kpt.pdf), and by Anwar and Fang (https://www.aeaweb.org/articles?id=10.1257/000282806776157579)
Thanks, I haven’t read them yet, but they look interesting.