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> Therefore, the Prosecutors Fallacy is a subtle error that requires careful thought to understand, which makes teaching it all the more challenging. Visual exp

by 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
The prosecutor's fallacy is the mistake of confusing the probability of a piece of evidence given guilt with the probability of guilt given a piece of evidence. In other words, it’s asking: “If Bob is guilty of this robbery, what are the chances we'd find his fingerprints at the crime scene?” When you should be asking: “Given that we've found Bob’s fingerprint at the crime scene, what’s the likelihood that he is guilty?"
- anonymous_sorry 3y ago
  > “If Bob is guilty of this robbery, what are the chances we'd find his fingerprints at the crime scene?” It's more like saying "the chance of these fingerprints being mistaken for Bob's are 1 in 100000. Therefore there's a 99999 in 100000 Bob is guilty of this robbery. Does Bob have a motive? Means? A history of crime? Is there evidence he was in the area? Could he have been in the house for another reason? And the biggie: did he only come under suspicion after police trawled through a database of 100000 fingerprints and matched the prints to Bob? Because you should expect a false positive in a database of that size.
  - cainxinth 3y ago
    You're right, mine misses the base rate part.
DSMan195276 3y ago
It's not quite a single sentence, but IMO the best example of the prosecutors fallacy is the lottery: The chance of winning the powerball lottery is 300 million to 1, so clearly anybody who wins _must_ have cheated since 300 million to 1 is practically impossible. Obviously that's not true, people legitimately win the powerball all the time despite the odds. The issue with this logic is judging the odds from the position of a _particular_ person winning vs. the odds of _anybody_ winning. Your individual odds are 1 in 300 million, but since the powerball gets many 100s of millions of entries clearly it's likely that eventually someone will win. Effectively, the prosecutors fallacy is presenting a very low probability as though that indicates something couldn't have happened. A probability alone is useless without the size of the population that probability is picking from, "low probability" events like winning the lottery happen all the time when your population is large enough.
- joe_the_user 3y ago
  That's a good explanation. The other to add is that someone combs a data file for "unlikely events" and finds one event they consider unlikely happened to a given person, you have to consider the probability of any event that looks unlikely occurring at all, not just the probability of that unlikely event happening.
- eep_social 3y ago
  Great explanation. I occasionally get to remind coworkers that at 100 requests per second, one in a million is about 8 times per day [1]. Usually mentioned in the context of handling, understanding, or planning for corner cases. [1] 100rps*60*60*24/1E6=8.64
- cbsmith 3y ago
  I often take it to the ridiculous point and use a scenario with only one possible positive case. Like: There is only one person in the world who is you. Let's say I had a facial recognition system that had a false positive rate of only 1 in a million. If I tried to find you by having the system scan through pictures of literally everyone on earth, how many people would I incorrectly identify as you?
aidenn0 3y ago
I tried several times to make it fit in one sentence, but short of a Dickensian abuse of semi-colons it's not really possible to fully explain in a sentence. My best effort: If you have a very large imbalance in your population vs. your target group (e.g. everybody in NYC, vs the one person who robbed a bank, or the whole population vs. people with a rare disease), then seemingly strong evidence is much weaker than it appears. Longer example: Lets say we've matched the DNA of a bank robber to someone using a test with a 1-in-a-million false positive rate. That, on its own, means there is still over a 95% chance (23/24) that they are innocent given that with 24M people in the NYC area on a given day, we would expect 24 people to match. The prosecutors fallacy would be saying that there is instead only a 1-in-a-million chance that this person is innocent.
yoz 3y ago
The Prosecutor's Fallacy happens when someone misinterprets the low odds of a specific innocent circumstance as sufficient to show that the accused's innocence is equally unlikely, because they have implicitly assumed that the odds of guilt are higher. Basic example: take a deck of cards and throw them in the air so the cards land in a random arrangement. If someone looks at the arrangement and says, "the odds against these cards landing in this specific arrangement are trillions to one, therefore someone must have arranged them like this deliberately," that's the Prosecutor's Fallacy. For a tragic example from real life, see the case of Sally Clark[1] who was wrongly convicted for the murder of her two baby sons. At trial, the only other possible explanation given for their deaths was Sudden Infant Death Syndrome (SIDS). A paediatrician testified that the odds of a single baby dying of SIDS were 1 in 8,500, so the probability of two specific babies dying in this way was the square of those odds (1 in 72 million). He concluded that the high improbability of this specific innocent circumstance meant that murder was a more likely explanation. There were several other problems with this testimony (not to mention the evidence presented), but this demonstrates the Prosecutor's Fallacy: the paediatrician misinterpreted the odds of two specific babies dying from SIDS as the odds of ever seeing SIDS claim a pair of babies. The odds he quoted are in the millions, but the number of two-child families is in the millions too; this means that such events will probably happen eventually, thanks to the Law of Truly Large Numbers[2]. [1] https://en.wikipedia.org/wiki/Sally_Clark https://en.wikipedia.org/wiki/Sally_Clark [2] https://en.wikipedia.org/wiki/Law_of_truly_large_numbers https://en.wikipedia.org/wiki/Law_of_truly_large_numbers
- cbsmith 3y ago
  It's worse than that, because the 1 in 72 million calculation assumed that the contributing factors for SIDS are uniform across all children, when even then we knew absolutely that they were not. The UK, with a population of 67 million (and that's current population, not the population when she was on trial), has far fewer than 72 million households in the UK with two children, if the probability of SIDS was distributed uniformly over all children, the odds of there being a household where two kids died of SIDS in the UK at any moment in time would seem to be less likely than not. However, we knew even then that if one baby from a particular pair of parents and a particular environment dies of SIDS, the odds of a second baby dying from SIDS with the same parents & environment increase dramatically. So the odds of two babies in the same household dying from SIDS aren't nearly 1 in 72 million. The Prosecutor's Fallacy just makes the magnitude of the statistical miscalculation even more absurd. Sadly, everyone, including her own defense team, thought that the correct "expert witness" for this kind of testimony would be a medical expert, rather than a statistical expert.
cbsmith 3y ago
The clear explanation to compute the odds you need to combine the distribution of positives & negatives with the false positive (or false negative) rate; a probability derived purely from a false positive rate presumes a uniform distribution of positives and negatives.
torstenvl 3y ago
Imagine I have a simple electrical circuit on main power. This circuit has only one outlet on it, which is connected only to one floor lamp. Imagine further that the light is currently on. If I flip the switch off, what is the probability that the light turns off? If the light goes off, what is the probability that I flipped the switch off? Those probabilities are not identical. --- Imagine that I have been administered a standard FDA-approved HIV test. If I am HIV-negative, what is the likelihood that the test comes back positive? (I.e., what is the false positive rate?) If the test comes back positive, what is the likelihood that I am HIV-negative? Those probabilities are not identical. --- Imagine that your client's DNA is tested against DNA found at a murder scene, comparing the short tandem repeats at 13 locations. There is no other evidence linking your client to the crime scene or to the murder. If your client was not at the scene, what is the likelihood of all 13 loci being matches? If all 13 loci are matches, what is the likelihood your client was not at the scene? Those probabilities are not identical.

by xpe 3y ago

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

The prosecutor's fallacy is the mistake of confusing the probability of a piece of evidence given guilt with the probability of guilt given a piece of evidence. In other words, it’s asking: “If Bob is guilty of this robbery, what are the chances we'd find his fingerprints at the crime scene?” When you should be asking: “Given that we've found Bob’s fingerprint at the crime scene, what’s the likelihood that he is guilty?"

anonymous_sorry 3y ago
> “If Bob is guilty of this robbery, what are the chances we'd find his fingerprints at the crime scene?” It's more like saying "the chance of these fingerprints being mistaken for Bob's are 1 in 100000. Therefore there's a 99999 in 100000 Bob is guilty of this robbery. Does Bob have a motive? Means? A history of crime? Is there evidence he was in the area? Could he have been in the house for another reason? And the biggie: did he only come under suspicion after police trawled through a database of 100000 fingerprints and matched the prints to Bob? Because you should expect a false positive in a database of that size.
- cainxinth 3y ago
  You're right, mine misses the base rate part.

DSMan195276 3y ago

It's not quite a single sentence, but IMO the best example of the prosecutors fallacy is the lottery: The chance of winning the powerball lottery is 300 million to 1, so clearly anybody who wins _must_ have cheated since 300 million to 1 is practically impossible. Obviously that's not true, people legitimately win the powerball all the time despite the odds. The issue with this logic is judging the odds from the position of a _particular_ person winning vs. the odds of _anybody_ winning. Your individual odds are 1 in 300 million, but since the powerball gets many 100s of millions of entries clearly it's likely that eventually someone will win. Effectively, the prosecutors fallacy is presenting a very low probability as though that indicates something couldn't have happened. A probability alone is useless without the size of the population that probability is picking from, "low probability" events like winning the lottery happen all the time when your population is large enough.

joe_the_user 3y ago
That's a good explanation. The other to add is that someone combs a data file for "unlikely events" and finds one event they consider unlikely happened to a given person, you have to consider the probability of any event that looks unlikely occurring at all, not just the probability of that unlikely event happening.
eep_social 3y ago
Great explanation. I occasionally get to remind coworkers that at 100 requests per second, one in a million is about 8 times per day [1]. Usually mentioned in the context of handling, understanding, or planning for corner cases. [1] 100rps*60*60*24/1E6=8.64
cbsmith 3y ago
I often take it to the ridiculous point and use a scenario with only one possible positive case. Like: There is only one person in the world who is you. Let's say I had a facial recognition system that had a false positive rate of only 1 in a million. If I tried to find you by having the system scan through pictures of literally everyone on earth, how many people would I incorrectly identify as you?

aidenn0 3y ago

I tried several times to make it fit in one sentence, but short of a Dickensian abuse of semi-colons it's not really possible to fully explain in a sentence. My best effort: If you have a very large imbalance in your population vs. your target group (e.g. everybody in NYC, vs the one person who robbed a bank, or the whole population vs. people with a rare disease), then seemingly strong evidence is much weaker than it appears. Longer example: Lets say we've matched the DNA of a bank robber to someone using a test with a 1-in-a-million false positive rate. That, on its own, means there is still over a 95% chance (23/24) that they are innocent given that with 24M people in the NYC area on a given day, we would expect 24 people to match. The prosecutors fallacy would be saying that there is instead only a 1-in-a-million chance that this person is innocent.

yoz 3y ago

The Prosecutor's Fallacy happens when someone misinterprets the low odds of a specific innocent circumstance as sufficient to show that the accused's innocence is equally unlikely, because they have implicitly assumed that the odds of guilt are higher. Basic example: take a deck of cards and throw them in the air so the cards land in a random arrangement. If someone looks at the arrangement and says, "the odds against these cards landing in this specific arrangement are trillions to one, therefore someone must have arranged them like this deliberately," that's the Prosecutor's Fallacy. For a tragic example from real life, see the case of Sally Clark[1] who was wrongly convicted for the murder of her two baby sons. At trial, the only other possible explanation given for their deaths was Sudden Infant Death Syndrome (SIDS). A paediatrician testified that the odds of a single baby dying of SIDS were 1 in 8,500, so the probability of two specific babies dying in this way was the square of those odds (1 in 72 million). He concluded that the high improbability of this specific innocent circumstance meant that murder was a more likely explanation. There were several other problems with this testimony (not to mention the evidence presented), but this demonstrates the Prosecutor's Fallacy: the paediatrician misinterpreted the odds of two specific babies dying from SIDS as the odds of ever seeing SIDS claim a pair of babies. The odds he quoted are in the millions, but the number of two-child families is in the millions too; this means that such events will probably happen eventually, thanks to the Law of Truly Large Numbers[2]. [1] https://en.wikipedia.org/wiki/Sally_Clark https://en.wikipedia.org/wiki/Sally_Clark [2] https://en.wikipedia.org/wiki/Law_of_truly_large_numbers https://en.wikipedia.org/wiki/Law_of_truly_large_numbers

cbsmith 3y ago
It's worse than that, because the 1 in 72 million calculation assumed that the contributing factors for SIDS are uniform across all children, when even then we knew absolutely that they were not. The UK, with a population of 67 million (and that's current population, not the population when she was on trial), has far fewer than 72 million households in the UK with two children, if the probability of SIDS was distributed uniformly over all children, the odds of there being a household where two kids died of SIDS in the UK at any moment in time would seem to be less likely than not. However, we knew even then that if one baby from a particular pair of parents and a particular environment dies of SIDS, the odds of a second baby dying from SIDS with the same parents & environment increase dramatically. So the odds of two babies in the same household dying from SIDS aren't nearly 1 in 72 million. The Prosecutor's Fallacy just makes the magnitude of the statistical miscalculation even more absurd. Sadly, everyone, including her own defense team, thought that the correct "expert witness" for this kind of testimony would be a medical expert, rather than a statistical expert.

cbsmith 3y ago

The clear explanation to compute the odds you need to combine the distribution of positives & negatives with the false positive (or false negative) rate; a probability derived purely from a false positive rate presumes a uniform distribution of positives and negatives.

torstenvl 3y ago

Imagine I have a simple electrical circuit on main power. This circuit has only one outlet on it, which is connected only to one floor lamp. Imagine further that the light is currently on. If I flip the switch off, what is the probability that the light turns off? If the light goes off, what is the probability that I flipped the switch off? Those probabilities are not identical. --- Imagine that I have been administered a standard FDA-approved HIV test. If I am HIV-negative, what is the likelihood that the test comes back positive? (I.e., what is the false positive rate?) If the test comes back positive, what is the likelihood that I am HIV-negative? Those probabilities are not identical. --- Imagine that your client's DNA is tested against DNA found at a murder scene, comparing the short tandem repeats at 13 locations. There is no other evidence linking your client to the crime scene or to the murder. If your client was not at the scene, what is the likelihood of all 13 loci being matches? If all 13 loci are matches, what is the likelihood your client was not at the scene? Those probabilities are not identical.