6 ms·

The Prosecutor's Fallacy (2018)

taneq 3y ago
This seems to me to be the equivalent of begging the question ('assuming the consequent', not 'requesting the question be asked').
- mannykannot 3y ago
  I think there are many examples where no question-begging premises are involved. For example, in the somewhat canonical example of incorrectly inferring the presence of a disease from a test when the base rate is low [1], the premises are the positive test result, the false-positive and false-negative rates for the test, and whatever premises about statistical reasoning lead to the calculation of the incorrect probability. [1] https://en.wikipedia.org/wiki/Base_rate_fallacy#Example_1:_Disease https://en.wikipedia.org/wiki/Base_rate_fallacy#Example_1:_D...
derbOac 3y ago
I wish these types of fallacies, related to ignoring base rates and equating likelihoods with posteriors, were more widely appreciated. However, a big problem in practice — and a reason why these fallacies exist aside from simple ignorance or mistake — is that the correct number, the posterior, requires knowing the prior base rates of something, which is often unknown. In some settings, the base rates are very well-characterized, but in others you really have no idea. In a lot of those cases, knowing the prior is qualitatively similar to knowing the posterior, which you're trying to figure out, so all you're left with with any certainty is the likelihood. The fallacy exists, it's important, but sometimes I think there's a bit more to it than simple ignorance. Sometimes there's no information on prior rates, or you don't really know which prior distribution something comes from, there's a mixture for example.
- makeitdouble 3y ago
  I feel it could come down to not using statistics to infer a given conclusion. For instance base probability accounted, if there was a 1 in a trillion chance someone was at the right place the right time, just straight assuming it couldn't happen by chance is still wrong. By definition that chance was not 0. At some point a practical decision could be made to cut prosecution cost, but it should be understood that nothing was proven.
  - RandomLensman 3y ago
    And if that even happens a lot in the world (e.g., many nurses looking at many patients every day), then that chance is quite likely to realize somewhere.
  - wongarsu 3y ago
    I think the legal system understands quite well that it doesn't prove anything (in the mathematical sense), which is why it has different standards of proof. A 1 in 1 trillion chance would be considered both "beyond reasonable doubt" (enough for criminal matters) and satisfy the much weaker "balance of probabilities" usually applied to civil matters. Of course there are plenty of examples to point to where people were convicted despite very reasonable doubt.
- nonameiguess 3y ago
  I've never actually heard it called the "prosecutor's fallacy" before, but it should be obvious why. The sort of thing often being looked for is something like "is planning an insurrection against the government," "is a terrorist," "is a murderer." We don't know the true base rates for any of these things, but we do know the base rate is very low. Almost nobody is a terrorist or a murderer. Also, it becomes easier to understand some types of frustrating or often wrong processes when we take into accounts not just rates of different error types, but the relative costs of them. A whole lot of criminals never see justice or get off on technicalities because the social cost in terms of destroying trust in the legal system is much higher for putting innocent people in prison than it is for failing to catch or failing to punish the guilty. Why do we seem to go so overboard with cancer screenings when the false positive rate is so high? Because the worst that can happen is mostly annoyance, wasted time, and wasted money. The worst that can happen with false negatives is you die. Why do our dragnets for terrorists seem to be so much more sensitive than dragnets for murder, rape, and property crime? Because even though the false positive rate here is even higher, the damage done by a successful terrorist attack is far greater. Why are FAANG hiring processes so jacked up? Because, right or wrong, the cost of hiring a bad engineer is perceived to be far higher than the cost of failing to hire a good one, especially when you get so many applicants that you're guaranteed to staff to the level you need virtually no matter how high a rate you reject at.
  - denton-scratch 3y ago
    > the damage done by a successful terrorist attack is far greater. Terrorist attacks are incredibly rare; murder, rape and (especially) property crime are commonplace, and don't rate even a column-inch in newspapers. Once upon a time, kids, there was a job called "court reporter"). How many people have you known who were victims of terrorist attacks? Right - zero. I don't know anyone who knows anyone who was the victim of a terrorist attack. How many people do you know who haven't been the victim of a personal crime, like assault, robbery or rape? Again, the expected answer is roughly zero. Most successful applications of political violence (I hate the term "terrorism") result in just a handful of deaths/injuries. By "successful" I don't mean they achieved their objectives; I just mean the attack wasn't foiled before it happened.
  - 3y ago
kmm 3y ago
I find Bayes' theorem is a bit more intuitive when written out not for probabilities but for the odds ratios, i.e. the ratio of the probabilities of something being true and something being false. Quite often one of these two will be close to zero, meaning the odds ratio will be close to (the inverse of) the probability anyway. So, letting H be the hypothesis and E the evidence, Bayes' theorem looks like o(H|E) = o(H) P(E|H)/P(E|¬H) Or in words, you update the odds ratio simply by multiplying it with the ratio of likelihoods of observing the evidence. Because odds ratios range through all positive numbers, you obviate the annoying normalization step you need with probabilities, which need to remain between 0 and 1. Applied to the example in the article, a test with with a false positive ratio of 1% can only ever boost the odds ratio of your believe by a factor of 100 (and in the case of the article it's even only 98:1), so if a disease has a prevalence of about 2:10000, such a test can only bring it to about 2:100.
- mtklein 3y ago
  On the topic of finding Bayes' theorem intuitively, I can never remember it on its own, but starting from the joint probability P(A,B) for any arbitrary A and B always helps me: // A and B are arbitrary. P(A,B) = P(B,A) // These two make sense if I sound them out in words. P(A,B) = P(A) P(B|A) P(B,A) = P(B) P(A|B) // Combine to give Bayes' theorem. P(A) P(B|A) = P(B) P(A|B)
  - n4r9 3y ago
    Exactly! The mathematical aspect of Bayes' Theorem is a simple algebraic manipulation of conditional probabilities. Another way to intuitively justify your middle two equations is to visualise a Venn diagram with overlapping circles representing A and B, such that areas correspond to probabilities. Then P(B|A) is the area of the overlap - i.e. P(A,B) - divided by the area of A - i.e. P(A).
  - mananaysiempre 3y ago
    The advantage of the odds formulation is that you can compute posterior odds in your head, whereas computing the posterior probabilities via the standard form of the theorem (that you wrote down here) involves an unpleasant normalization factor. E.g. for the example in https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidence_population https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidenc...: prior odds of infection = 2 : 98 × likelihood ratio = 100 : 5 = posterior odds ≈ 200 : 500 and thus the probabilities (if you actually need them at this point) are ≈ (2/7, 5/7). See also 3Blue1Brown’s exposition: https://youtu.be/watch?v=lG4VkPoG3ko https://youtu.be/watch?v=lG4VkPoG3ko.
thechao 3y ago
This fallacy always reminds me of the "Jonah" sequence in the "Master and Commander" movie; or, any of the witch tests used by Monty Python.
xg15 3y ago
Another way to illustrate the base rate fallacy is to get rid of any randomness for a moment and imagine a perfectly regular and deterministic toy universe: For the disease test, suppose every patient has a unique ID, starting at #0 and counting up consecutively. Because it's a toy universe, we can say that the actually infected patients are exactly the ones with IDs #0, #10000, #20000 etc. The test will come back positive if either the patient is infected (i.e. false-negative rate is 0, P(positive|infected) is 1.0) or if the ID is one of #1, #1001, #2001, #3001, etc. This results in a base rate of 1/10000 and a false positive rate of (almost) 1/1000. If you now look at all the IDs for which the test will be positive, those will be: #0, #1, #1001, #2001, ..., #9001, #10000, #10001, #11001, #12001, ..., #20000, #20001, #21001, etc etc. It's easy to see that for each true positive (IDs that end with 0), there are 10 false positives (IDs that end with 1) - so the probability that some specific ID from that list is a true positive P(infected|positive) is 1/10 = 0.1, i.e. it's more likely the patient is not infected than that they are. (It's still more likely that they are infected than it would be for a random pick from the general population: 1/10 vs 1/10000) Now consider a second population with a higher base rate: Now the infected patients are #0, #50, #100, #150 etc, i.e. the base rate is 1/50. If you look at the IDs with positive tests again, you get #0, #1, #50, #100, #150, ..., #1000, #1001, #1050, ..., #2000, #2001 etc etc. Now there are 50 true positives for each false positive, i.e. P(infected|positive) is something like 50/51 =~ 0.98. So the test suddenly got much more "reliable", even though the false-positive rate didn't change, only through a change in base rate.
1970-01-01 3y ago
>He mustn’t love me anymore, as it’s been 3 days and he hasn’t returned my call. I understand the others, but what's the fallacy here? Assuming all return calls (evidence) must occur within X days?
- hammock 3y ago
  It’s likely he’s just busy
  - deleted 3y ago
    [deleted]
  - lr4444lr 3y ago
    ... and still, he doesn't find talking to her relaxing at the end of the day. Unless he is incapacitated or dead, definitely a huge red flag.
- nonameiguess 3y ago
  It's the same as all the others. Consider the base rate for all possible reasons a person might not return a call: - Has been injured or incapacitated - Didn't see the call - Didn't receive the call - He actually did return the call, but you didn't notice or didn't receive it - Has been swamped by other demands - Simply forgot - Lost his phone - Has become generally depressed or despondent - Fell out of love with the caller and is avoiding saying so It doesn't seem likely that the rate for the last reason is high enough to outweigh all of the other possible reasons, and the evidence you have is equal evidence for all possible causes, not evidence specifically that he stopped loving you.
  - xpe 3y ago
    A nice uplifting list indeed. As to all save the last two, one might hope that eventually love would manifest as action. Emotions, logic, and time are complicated for people.
  - denton-scratch 3y ago
    > not evidence specifically that he stopped loving you. I think that last example is problematic. Did he ever start loving you? How would you know? How would he know? What is love, anyway? (~Charles Rex) What evidence would support the belief that someone does or doesn't love you? Erich Fromm defined four types of love; but just because someone is your mother doesn't mean that their attitude is one of Motherly Love. They might not even like you. Basically, "Love" is a field with no definitions, no evidence, and nothing even close to certainty. It's all just feels.
xpe 3y ago
> Therefore, the Prosecutors Fallacy is a subtle error that requires careful thought to understand, which makes teaching it all the more challenging. Visual explanations may be helpful. Indeed. I've read and re-read the article. I don't see a clear explanation of the fallacy that fits into one sentence.
- anonymous_sorry 3y ago
  "False positives are rare" does not imply that a positive result it is probably true. True positives might be even rarer than false positives. The problem is false positive rates are usually expressed in terms of "what proportion of negative cases will be reported as positive?" which can't answer the question "what proportion of positive results will be wrong?". Answering the latter also depends on how many true positives there are, which may vary, or even be completely unknown. The false negative rate also needs to be factored in.
  - cbsmith 3y ago
    Yup. It's very similar to the problem of explaining the importance of precision and recall in information retrieval.
- Spooky23 3y ago
  It's pretty simple, but in trying to demonstrate why, the author misses the point: When an individual with something to gain asserts some probabilities, disregard that assertion. Forget about medicine or criminal trials. Think of a sales professional - their assertions are commonly disregarded (rightly) as fluff. But put a white coat or nice suit on and people get deferential. A prosecutor in the courtroom is little different than a car salesman.
  - anonymous_sorry 3y ago
    That is absolutely not the prosecutor's fallacy. What you're describing is closer to "argument from authority".
- cainxinth 3y ago
xpe 3y ago
TLDR: I highly recommend HN readers skip this poorly-written article and go read "Example 2: Drunk drivers" on Wikipedia's article about the Base Rate Fallacy instead [1]. You'll probably learn it more and get better context. [1] https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy instead. Ok, I'm going to put on my harsh editor's hat. I'll abbreviate the Prosecutor's Fallacy as PF. Here are some glaring deficiencies: 1. The article goes not get to the point quickly (or at all!); it doesn't define PF up front nor ever. 2. The article doesn't properly situate the concept; it does not recognize synonyms for PF, which include: "base rate fallacy", "base rate neglect", or "base rate bias", "defense attorney's fallacy". 3. Poor logical flow. One example: Saying the PF "involves" conditional probability without having defined it first, much less at all -- is frustrating to the reader. 4. Poor reasoning. One example: The claim that the PF "is most often associated with miscarriages of justice." isn't plausible nor defended. It should be rephrased to say "is often associated". 5. A lack of organizational cues. Most obviously, the article doesn't have any headings. It desperately needs them. 6. Various formatting problems. For one, the top quotation should be formatted as a blockquote. Overall, the article would benefit from several more drafts. I'm pretty disappointed a professional organization would hit "publish" on this one. I'll try to find a way to share my feedback with the editor(s) and author. In the meanwhile, I suggest HN people read this instead: https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy Why? The concept is explained in a self-contained way, followed by self-contained examples. I really appreciate that approach. Slight tangent: On a more happy note, I think a lot of software developers are better writers than they give themselves credit for. I don't think most software developers would make the same mistakes as this author. I'm not saying we're all great writers, but many of us do have a strong sense of logical flow and organization.
EGreg 3y ago
Conditional probabilities are what we actually operate with in the real world. They are all independent — in a chain of events, conditioning B on A makes the result independent of A. Like a sales funnel, each step can be multiplied because they are conditional that you got up to thay point!
themgt 3y ago
One example of this is people saying "you have a 1 in 10 million chance of getting attacked by a shark!" I thought about this swimming for an hour+ a day at a beach with a huge amount of sharks. Like "1 in 10 million? What's the average number of hours per year per American spent full body swimming in shark infested waters? Not even a very high % of people who live in this area spend as much time doing that as I do. If I spend 500 hours per year doing that, my risk calculation has gotta be about 50,000 times worse than average."
- NotYourLawyer 3y ago
  https://xkcd.com/795/ https://xkcd.com/795/
- joe_the_user 3y ago
  But even more, some number of shark bites occur each year and for the people bitten, the probability of having been bitten is 1. The main thing is that sure, "the probability of this highly incriminating series of events happening to you without you being guilty" may be very small. But probability of "A series of highly incriminating and unlikely series of events happening somewhere, to someone" may be very high, nearly one.
  - hoseja 3y ago
    >for the people bitten, the probability of having been bitten is 1 That's completely contrary to the usual use of probability. The probability of having been bitten for everyone else is then 0 and the concept becomes meaningless.
kr0bat 3y ago
>Notice that at 0.02% prevalence the two conditional probabilities differ by 97% but at 20% prevalence the difference is only 3%. Therefore, the Prosecutor’s Fallacy is not an issue when the prevalence (or prior likelihood of guilt) is high, because the conditional probabilities are similar. Whoa I think the point makes sense but I'm not sure about the data used to demonstrate it. The difference between probabilities at 20% prevalence were 1% and 4%. That's not a 3% difference, that's a 3 percentage point difference that results in a 400% difference in probability. That's not similar at all.
deleted 3y ago
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igiveup 3y ago
Isn't this just hypothesis testing? [1] Zero hypothesis: N deaths happen by chance, given a known probability distribution of patient deaths. Alternative: There was a different probability distribution in play (apparently facilitated by a specific nurse). P-value: 1 in 342 million, really convincing. So, is the fallacy that somebody calls the number "probability" rather than "p-value"? Or am I getting it wrong? [1] https://en.wikipedia.org/wiki/Statistical_hypothesis_testing https://en.wikipedia.org/wiki/Statistical_hypothesis_testing
- sealeck 3y ago
  No the issue is when prosecutors mix up Pr[evidence | innocence] with Pr[innocence | evidence]. It isn't correct to conclude from the former that someone is guilty.
  - igiveup 3y ago
    I believe Pr[evidence | innocence] is p-value (or maybe one minus p-value, not sure). Statisticians use this routinely to test "innocence". It does not mean probability of innocence, but it means something.
    - hgomersall 3y ago
      Yeah, it means the probability of the evidence given the person is innocent. If they use it for anything else they're using it wrongly and they shouldn't.
- ivanbakel 3y ago
  No, the fallacy comes from choosing the sample before analysing the probability. The probability a specific nurse could have such specific bad luck is very low, but there are of course many nurses, and each nurse treats many patients. What is the probability any nurse would have such bad luck, over a long period? How does that probability compare to the probability of murder, which is also estimable? Only either unlucky nurses or murderers end up in the docket - so the p-value really depends on the probability that the prosecutor faces an unlucky nurse versus a murderer. A simpler comparison: a die with a thousand faces is quite unlikely to land on any particular face. When you roll it, it gives you a sample - is it more likely that the die is weighted to that face, or that the die is fair?
chmod600 3y ago
Intellectually I understand this, but it's really really hard (for me) to consistently avoid this fallacy.
cbsmith 3y ago
I always explain it this way: "The odds of getting a Royal Flush in poker are 649739 : 1. So, if you are playing someone and they present a Royal Flush, what are the odds they are cheating? What if the person you were playing was the Dali Lama [or insert another person who has little reason to cheat]?"
shib71 3y ago
Reminds me of an old Discworld chestnut: Scientists have calculated that the chances of something so patently absurd actually existing are millions to one. But magicians have calculated that million-to-one chances crop up nine times out of ten.
chongli 3y ago
I have to say that this is one of the most frustrating things about the institution of criminal law. Knowledge of Bayes’ theorem, and statistics more broadly, already exists within our culture. Yet this fallacy continues to be applied in court, leading to horrific miscarriages of justice. Why can’t we just solve this problem? Educate judges in basic statistics, appoint court statisticians for more complex cases, and allow judges to hold people in contempt of court for trying to make these kinds of statistically fallacious arguments.

This seems to me to be the equivalent of begging the question ('assuming the consequent', not 'requesting the question be asked').

mannykannot 3y ago
I think there are many examples where no question-begging premises are involved. For example, in the somewhat canonical example of incorrectly inferring the presence of a disease from a test when the base rate is low [1], the premises are the positive test result, the false-positive and false-negative rates for the test, and whatever premises about statistical reasoning lead to the calculation of the incorrect probability. [1] https://en.wikipedia.org/wiki/Base_rate_fallacy#Example_1:_Disease https://en.wikipedia.org/wiki/Base_rate_fallacy#Example_1:_D...

derbOac 3y ago

I wish these types of fallacies, related to ignoring base rates and equating likelihoods with posteriors, were more widely appreciated. However, a big problem in practice — and a reason why these fallacies exist aside from simple ignorance or mistake — is that the correct number, the posterior, requires knowing the prior base rates of something, which is often unknown. In some settings, the base rates are very well-characterized, but in others you really have no idea. In a lot of those cases, knowing the prior is qualitatively similar to knowing the posterior, which you're trying to figure out, so all you're left with with any certainty is the likelihood. The fallacy exists, it's important, but sometimes I think there's a bit more to it than simple ignorance. Sometimes there's no information on prior rates, or you don't really know which prior distribution something comes from, there's a mixture for example.

makeitdouble 3y ago
I feel it could come down to not using statistics to infer a given conclusion. For instance base probability accounted, if there was a 1 in a trillion chance someone was at the right place the right time, just straight assuming it couldn't happen by chance is still wrong. By definition that chance was not 0. At some point a practical decision could be made to cut prosecution cost, but it should be understood that nothing was proven.
- RandomLensman 3y ago
  And if that even happens a lot in the world (e.g., many nurses looking at many patients every day), then that chance is quite likely to realize somewhere.
- wongarsu 3y ago
  I think the legal system understands quite well that it doesn't prove anything (in the mathematical sense), which is why it has different standards of proof. A 1 in 1 trillion chance would be considered both "beyond reasonable doubt" (enough for criminal matters) and satisfy the much weaker "balance of probabilities" usually applied to civil matters. Of course there are plenty of examples to point to where people were convicted despite very reasonable doubt.
nonameiguess 3y ago
I've never actually heard it called the "prosecutor's fallacy" before, but it should be obvious why. The sort of thing often being looked for is something like "is planning an insurrection against the government," "is a terrorist," "is a murderer." We don't know the true base rates for any of these things, but we do know the base rate is very low. Almost nobody is a terrorist or a murderer. Also, it becomes easier to understand some types of frustrating or often wrong processes when we take into accounts not just rates of different error types, but the relative costs of them. A whole lot of criminals never see justice or get off on technicalities because the social cost in terms of destroying trust in the legal system is much higher for putting innocent people in prison than it is for failing to catch or failing to punish the guilty. Why do we seem to go so overboard with cancer screenings when the false positive rate is so high? Because the worst that can happen is mostly annoyance, wasted time, and wasted money. The worst that can happen with false negatives is you die. Why do our dragnets for terrorists seem to be so much more sensitive than dragnets for murder, rape, and property crime? Because even though the false positive rate here is even higher, the damage done by a successful terrorist attack is far greater. Why are FAANG hiring processes so jacked up? Because, right or wrong, the cost of hiring a bad engineer is perceived to be far higher than the cost of failing to hire a good one, especially when you get so many applicants that you're guaranteed to staff to the level you need virtually no matter how high a rate you reject at.
- denton-scratch 3y ago
  > the damage done by a successful terrorist attack is far greater. Terrorist attacks are incredibly rare; murder, rape and (especially) property crime are commonplace, and don't rate even a column-inch in newspapers. Once upon a time, kids, there was a job called "court reporter"). How many people have you known who were victims of terrorist attacks? Right - zero. I don't know anyone who knows anyone who was the victim of a terrorist attack. How many people do you know who haven't been the victim of a personal crime, like assault, robbery or rape? Again, the expected answer is roughly zero. Most successful applications of political violence (I hate the term "terrorism") result in just a handful of deaths/injuries. By "successful" I don't mean they achieved their objectives; I just mean the attack wasn't foiled before it happened.
- 3y ago

kmm 3y ago

I find Bayes' theorem is a bit more intuitive when written out not for probabilities but for the odds ratios, i.e. the ratio of the probabilities of something being true and something being false. Quite often one of these two will be close to zero, meaning the odds ratio will be close to (the inverse of) the probability anyway. So, letting H be the hypothesis and E the evidence, Bayes' theorem looks like o(H|E) = o(H) P(E|H)/P(E|¬H) Or in words, you update the odds ratio simply by multiplying it with the ratio of likelihoods of observing the evidence. Because odds ratios range through all positive numbers, you obviate the annoying normalization step you need with probabilities, which need to remain between 0 and 1. Applied to the example in the article, a test with with a false positive ratio of 1% can only ever boost the odds ratio of your believe by a factor of 100 (and in the case of the article it's even only 98:1), so if a disease has a prevalence of about 2:10000, such a test can only bring it to about 2:100.

mtklein 3y ago
On the topic of finding Bayes' theorem intuitively, I can never remember it on its own, but starting from the joint probability P(A,B) for any arbitrary A and B always helps me: // A and B are arbitrary. P(A,B) = P(B,A) // These two make sense if I sound them out in words. P(A,B) = P(A) P(B|A) P(B,A) = P(B) P(A|B) // Combine to give Bayes' theorem. P(A) P(B|A) = P(B) P(A|B)
- n4r9 3y ago
  Exactly! The mathematical aspect of Bayes' Theorem is a simple algebraic manipulation of conditional probabilities. Another way to intuitively justify your middle two equations is to visualise a Venn diagram with overlapping circles representing A and B, such that areas correspond to probabilities. Then P(B|A) is the area of the overlap - i.e. P(A,B) - divided by the area of A - i.e. P(A).
- mananaysiempre 3y ago
  The advantage of the odds formulation is that you can compute posterior odds in your head, whereas computing the posterior probabilities via the standard form of the theorem (that you wrote down here) involves an unpleasant normalization factor. E.g. for the example in https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidence_population https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidenc...: prior odds of infection = 2 : 98 × likelihood ratio = 100 : 5 = posterior odds ≈ 200 : 500 and thus the probabilities (if you actually need them at this point) are ≈ (2/7, 5/7). See also 3Blue1Brown’s exposition: https://youtu.be/watch?v=lG4VkPoG3ko https://youtu.be/watch?v=lG4VkPoG3ko.

thechao 3y ago

This fallacy always reminds me of the "Jonah" sequence in the "Master and Commander" movie; or, any of the witch tests used by Monty Python.

xg15 3y ago

Another way to illustrate the base rate fallacy is to get rid of any randomness for a moment and imagine a perfectly regular and deterministic toy universe: For the disease test, suppose every patient has a unique ID, starting at #0 and counting up consecutively. Because it's a toy universe, we can say that the actually infected patients are exactly the ones with IDs #0, #10000, #20000 etc. The test will come back positive if either the patient is infected (i.e. false-negative rate is 0, P(positive|infected) is 1.0) or if the ID is one of #1, #1001, #2001, #3001, etc. This results in a base rate of 1/10000 and a false positive rate of (almost) 1/1000. If you now look at all the IDs for which the test will be positive, those will be: #0, #1, #1001, #2001, ..., #9001, #10000, #10001, #11001, #12001, ..., #20000, #20001, #21001, etc etc. It's easy to see that for each true positive (IDs that end with 0), there are 10 false positives (IDs that end with 1) - so the probability that some specific ID from that list is a true positive P(infected|positive) is 1/10 = 0.1, i.e. it's more likely the patient is not infected than that they are. (It's still more likely that they are infected than it would be for a random pick from the general population: 1/10 vs 1/10000) Now consider a second population with a higher base rate: Now the infected patients are #0, #50, #100, #150 etc, i.e. the base rate is 1/50. If you look at the IDs with positive tests again, you get #0, #1, #50, #100, #150, ..., #1000, #1001, #1050, ..., #2000, #2001 etc etc. Now there are 50 true positives for each false positive, i.e. P(infected|positive) is something like 50/51 =~ 0.98. So the test suddenly got much more "reliable", even though the false-positive rate didn't change, only through a change in base rate.

1970-01-01 3y ago

>He mustn’t love me anymore, as it’s been 3 days and he hasn’t returned my call. I understand the others, but what's the fallacy here? Assuming all return calls (evidence) must occur within X days?

hammock 3y ago
It’s likely he’s just busy
- deleted 3y ago
  [deleted]
- lr4444lr 3y ago
  ... and still, he doesn't find talking to her relaxing at the end of the day. Unless he is incapacitated or dead, definitely a huge red flag.
nonameiguess 3y ago
It's the same as all the others. Consider the base rate for all possible reasons a person might not return a call: - Has been injured or incapacitated - Didn't see the call - Didn't receive the call - He actually did return the call, but you didn't notice or didn't receive it - Has been swamped by other demands - Simply forgot - Lost his phone - Has become generally depressed or despondent - Fell out of love with the caller and is avoiding saying so It doesn't seem likely that the rate for the last reason is high enough to outweigh all of the other possible reasons, and the evidence you have is equal evidence for all possible causes, not evidence specifically that he stopped loving you.
- xpe 3y ago
  A nice uplifting list indeed. As to all save the last two, one might hope that eventually love would manifest as action. Emotions, logic, and time are complicated for people.
- denton-scratch 3y ago
  > not evidence specifically that he stopped loving you. I think that last example is problematic. Did he ever start loving you? How would you know? How would he know? What is love, anyway? (~Charles Rex) What evidence would support the belief that someone does or doesn't love you? Erich Fromm defined four types of love; but just because someone is your mother doesn't mean that their attitude is one of Motherly Love. They might not even like you. Basically, "Love" is a field with no definitions, no evidence, and nothing even close to certainty. It's all just feels.

xpe 3y ago

> Therefore, the Prosecutors Fallacy is a subtle error that requires careful thought to understand, which makes teaching it all the more challenging. Visual explanations may be helpful. Indeed. I've read and re-read the article. I don't see a clear explanation of the fallacy that fits into one sentence.

anonymous_sorry 3y ago
"False positives are rare" does not imply that a positive result it is probably true. True positives might be even rarer than false positives. The problem is false positive rates are usually expressed in terms of "what proportion of negative cases will be reported as positive?" which can't answer the question "what proportion of positive results will be wrong?". Answering the latter also depends on how many true positives there are, which may vary, or even be completely unknown. The false negative rate also needs to be factored in.
- cbsmith 3y ago
  Yup. It's very similar to the problem of explaining the importance of precision and recall in information retrieval.
Spooky23 3y ago
It's pretty simple, but in trying to demonstrate why, the author misses the point: When an individual with something to gain asserts some probabilities, disregard that assertion. Forget about medicine or criminal trials. Think of a sales professional - their assertions are commonly disregarded (rightly) as fluff. But put a white coat or nice suit on and people get deferential. A prosecutor in the courtroom is little different than a car salesman.
- anonymous_sorry 3y ago
  That is absolutely not the prosecutor's fallacy. What you're describing is closer to "argument from authority".
cainxinth 3y ago

xpe 3y ago

TLDR: I highly recommend HN readers skip this poorly-written article and go read "Example 2: Drunk drivers" on Wikipedia's article about the Base Rate Fallacy instead [1]. You'll probably learn it more and get better context. [1] https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy instead. Ok, I'm going to put on my harsh editor's hat. I'll abbreviate the Prosecutor's Fallacy as PF. Here are some glaring deficiencies: 1. The article goes not get to the point quickly (or at all!); it doesn't define PF up front nor ever. 2. The article doesn't properly situate the concept; it does not recognize synonyms for PF, which include: "base rate fallacy", "base rate neglect", or "base rate bias", "defense attorney's fallacy". 3. Poor logical flow. One example: Saying the PF "involves" conditional probability without having defined it first, much less at all -- is frustrating to the reader. 4. Poor reasoning. One example: The claim that the PF "is most often associated with miscarriages of justice." isn't plausible nor defended. It should be rephrased to say "is often associated". 5. A lack of organizational cues. Most obviously, the article doesn't have any headings. It desperately needs them. 6. Various formatting problems. For one, the top quotation should be formatted as a blockquote. Overall, the article would benefit from several more drafts. I'm pretty disappointed a professional organization would hit "publish" on this one. I'll try to find a way to share my feedback with the editor(s) and author. In the meanwhile, I suggest HN people read this instead: https://en.wikipedia.org/wiki/Base_rate_fallacy https://en.wikipedia.org/wiki/Base_rate_fallacy Why? The concept is explained in a self-contained way, followed by self-contained examples. I really appreciate that approach. Slight tangent: On a more happy note, I think a lot of software developers are better writers than they give themselves credit for. I don't think most software developers would make the same mistakes as this author. I'm not saying we're all great writers, but many of us do have a strong sense of logical flow and organization.

EGreg 3y ago

Conditional probabilities are what we actually operate with in the real world. They are all independent — in a chain of events, conditioning B on A makes the result independent of A. Like a sales funnel, each step can be multiplied because they are conditional that you got up to thay point!

themgt 3y ago

One example of this is people saying "you have a 1 in 10 million chance of getting attacked by a shark!" I thought about this swimming for an hour+ a day at a beach with a huge amount of sharks. Like "1 in 10 million? What's the average number of hours per year per American spent full body swimming in shark infested waters? Not even a very high % of people who live in this area spend as much time doing that as I do. If I spend 500 hours per year doing that, my risk calculation has gotta be about 50,000 times worse than average."

NotYourLawyer 3y ago
https://xkcd.com/795/ https://xkcd.com/795/
joe_the_user 3y ago
But even more, some number of shark bites occur each year and for the people bitten, the probability of having been bitten is 1. The main thing is that sure, "the probability of this highly incriminating series of events happening to you without you being guilty" may be very small. But probability of "A series of highly incriminating and unlikely series of events happening somewhere, to someone" may be very high, nearly one.
- hoseja 3y ago
  >for the people bitten, the probability of having been bitten is 1 That's completely contrary to the usual use of probability. The probability of having been bitten for everyone else is then 0 and the concept becomes meaningless.

kr0bat 3y ago

>Notice that at 0.02% prevalence the two conditional probabilities differ by 97% but at 20% prevalence the difference is only 3%. Therefore, the Prosecutor’s Fallacy is not an issue when the prevalence (or prior likelihood of guilt) is high, because the conditional probabilities are similar. Whoa I think the point makes sense but I'm not sure about the data used to demonstrate it. The difference between probabilities at 20% prevalence were 1% and 4%. That's not a 3% difference, that's a 3 percentage point difference that results in a 400% difference in probability. That's not similar at all.

deleted 3y ago

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igiveup 3y ago

Isn't this just hypothesis testing? [1] Zero hypothesis: N deaths happen by chance, given a known probability distribution of patient deaths. Alternative: There was a different probability distribution in play (apparently facilitated by a specific nurse). P-value: 1 in 342 million, really convincing. So, is the fallacy that somebody calls the number "probability" rather than "p-value"? Or am I getting it wrong? [1] https://en.wikipedia.org/wiki/Statistical_hypothesis_testing https://en.wikipedia.org/wiki/Statistical_hypothesis_testing

sealeck 3y ago
No the issue is when prosecutors mix up Pr[evidence | innocence] with Pr[innocence | evidence]. It isn't correct to conclude from the former that someone is guilty.
- igiveup 3y ago
  I believe Pr[evidence | innocence] is p-value (or maybe one minus p-value, not sure). Statisticians use this routinely to test "innocence". It does not mean probability of innocence, but it means something.
  - hgomersall 3y ago
    Yeah, it means the probability of the evidence given the person is innocent. If they use it for anything else they're using it wrongly and they shouldn't.
ivanbakel 3y ago
No, the fallacy comes from choosing the sample before analysing the probability. The probability a specific nurse could have such specific bad luck is very low, but there are of course many nurses, and each nurse treats many patients. What is the probability any nurse would have such bad luck, over a long period? How does that probability compare to the probability of murder, which is also estimable? Only either unlucky nurses or murderers end up in the docket - so the p-value really depends on the probability that the prosecutor faces an unlucky nurse versus a murderer. A simpler comparison: a die with a thousand faces is quite unlikely to land on any particular face. When you roll it, it gives you a sample - is it more likely that the die is weighted to that face, or that the die is fair?

chmod600 3y ago

Intellectually I understand this, but it's really really hard (for me) to consistently avoid this fallacy.

cbsmith 3y ago

I always explain it this way: "The odds of getting a Royal Flush in poker are 649739 : 1. So, if you are playing someone and they present a Royal Flush, what are the odds they are cheating? What if the person you were playing was the Dali Lama [or insert another person who has little reason to cheat]?"

shib71 3y ago

Reminds me of an old Discworld chestnut: Scientists have calculated that the chances of something so patently absurd actually existing are millions to one. But magicians have calculated that million-to-one chances crop up nine times out of ten.

chongli 3y ago

I have to say that this is one of the most frustrating things about the institution of criminal law. Knowledge of Bayes’ theorem, and statistics more broadly, already exists within our culture. Yet this fallacy continues to be applied in court, leading to horrific miscarriages of justice. Why can’t we just solve this problem? Educate judges in basic statistics, appoint court statisticians for more complex cases, and allow judges to hold people in contempt of court for trying to make these kinds of statistically fallacious arguments.