|
|
#11
02-09-2007, 03:29 PM
|
|||
|
|||
Re: The Cab Problem
phzon,
How about if I put the problem this way: Suppose every night there's a hit and run accident, caused by the same cab every night, for 1000 nights. Every night a different person sees the accident. All the witnesses have this 80%/20% probability of being right/wrong with identifying the color. This is some eye-brain problem with the people of this town. Every day there's a "trial" with regard to the previous day's accident. Now I randomly put you in court to see the 465th trial. The witness says the car from yesterday's accident was blue (we all know that there are 15% blue cabs in town, as in the original problem). What would you say is the probability of the car being blue? 
|
|
#12
02-09-2007, 04:20 PM
|
|||
|
|||
|
Re: The Cab Problem
The above example seems kind of silly if it's the same color cab that is causing the accident every night. After 464 trials, look at the color of the car that has been chosen about 80% of the time, and that is the color of the car that continues to cause accidents.
Back to the original question, I just did some quick simulations in Excel to suggest the Bayesian reasoning is correct. Using 1000 "nights" with probability of 85% of it being a green car and 15% of being a blue car, I then had an observer pick the correct color 80% of the time and the wrong color 20% of the time. I then looked at (# of times blue car was claimed to be seen and it truly was the blue car)/(# of times blue car was claimed to be seen) and it was 40% which is right in line with 41%.
|
|
#13
02-09-2007, 04:40 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
The above example seems kind of silly if it's the same color cab that is causing the accident every night. After 464 trials, look at the color of the car that has been chosen about 80% of the time, and that is the color of the car that continues to cause accidents. [/ QUOTE ] Well obviously you don't know what were the result before you came in to see the 465th trial. However, please notice how you said yourself that if you do check the previous witnesses' testimonies, the correct answer is the one that was claimed by about 80% of them, in other words - there is 0.8 probability that the 465th witness is correct - so if he says "blue" there's 80% the cab was blue. [ QUOTE ] back to the original question, I just did some quick simulations in Excel to suggest the Bayesian reasoning is correct. Using 1000 "nights" with probability of 85% of it being a green car and 15% of being a blue car, I then had an observer pick the correct color 80% of the time and the wrong color 20% of the time. I then looked at (# of times blue car was claimed to be seen and it truly was the blue car)/(# of times blue car was claimed to be seen) and it was 40% which is right in line with 41%. [/ QUOTE ] I believe that this does not prove much, as it is simply going in a circle. Surely the Bayesian theorem solution to this problem is correct if you use the Bayesian theorem assumption to solve the problem. I don't see why you would need Excel to show that.
|
|
#14
02-09-2007, 04:48 PM
|
|||
|
|||
|
Re: The Cab Problem
I would say the argument attributed to Cohen is correct here and that Bayes is misapplied, as the application of Bayes inevitably discounts the accuracy figure for the witness. In other words it's contrary to fact.
|
|
#15
02-09-2007, 05:00 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
[ QUOTE ] I think alThor's point is spot on. Suppose the prosecutor brings the witness who says he saw a blue cab and an expert who testifies that under controlled tests the witness has the 80% accuracy. At that point the jurors will weigh the evidence according to the 80% accuracy. But suppose the Defense then brings an expert who testifies that 0% of the cabs running that day were Blue. Clearly the Jurors will take that evidence into account and conclude the witness was mistaken. [/ QUOTE ] In this point of the trial it simply seems that the witness and the expert testifying about his accuracy are both completely irrelevant. The cab was green since it couldn't have been blue. [ QUOTE ] But now suppose instead that the Defense brings an expert who testifies that only 1% of the cabs running that day were Blue. Should the jurors not also take that evidence into consideration to conclude there is a high likelihood the Witness is mistaken? Surely even Cohen would agree they should. But what if it's 2%. Or 3%. Or 15%? At what point does Cohen argue that such evidence becomes irrelevant? [/ QUOTE ] This is an interesting perspective. I might think that in a way, as strange as it might sound, the more blue cabs running at the time of the accident (up to some point), the more relevant is the witness. Cohen is probably exaggerating when arguing that the witness is all that matters, but maybe there's some way of assessing how relevant is the witness, as some function of the base rate? [/ QUOTE ] Well yea. There is a way of assessing it. It's called Bayes Theorum. PairTheBoard
|
|
#16
02-09-2007, 05:11 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
phzon, How about if I put the problem this way: Suppose every night there's a hit and run accident, caused by the same cab every night, for 1000 nights. Every night a different person sees the accident. All the witnesses have this 80%/20% probability of being right/wrong with identifying the color. This is some eye-brain problem with the people of this town. Every day there's a "trial" with regard to the previous day's accident. Now I randomly put you in court to see the 465th trial. The witness says the car from yesterday's accident was blue (we all know that there are 15% blue cabs in town, as in the original problem). What would you say is the probability of the car being blue? [/ QUOTE ] I always like to look for a hidden assumption in a flawed argument. I think your setup above points to the hidden assumption being made by Cohen. Just like in your setup where the same car is causing all the accidents, I think Cohen is essentially saying, look, it was a Blue Cab that caused the accident. I know it was a Blue Cab that caused the accident. Therefore, the fact that only 1% or 2% or 15% of the cabs on the road were Blue that night is irrelevant. But don't take my word for it. Take the word of the witness who saw it as Blue with 80% accuracy and be 80% confident that he was right. PairTheBoard
|
|
#17
02-09-2007, 05:12 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
[ QUOTE ] [ QUOTE ] I think alThor's point is spot on. Suppose the prosecutor brings the witness who says he saw a blue cab and an expert who testifies that under controlled tests the witness has the 80% accuracy. At that point the jurors will weigh the evidence according to the 80% accuracy. But suppose the Defense then brings an expert who testifies that 0% of the cabs running that day were Blue. Clearly the Jurors will take that evidence into account and conclude the witness was mistaken. [/ QUOTE ] In this point of the trial it simply seems that the witness and the expert testifying about his accuracy are both completely irrelevant. The cab was green since it couldn't have been blue. [ QUOTE ] But now suppose instead that the Defense brings an expert who testifies that only 1% of the cabs running that day were Blue. Should the jurors not also take that evidence into consideration to conclude there is a high likelihood the Witness is mistaken? Surely even Cohen would agree they should. But what if it's 2%. Or 3%. Or 15%? At what point does Cohen argue that such evidence becomes irrelevant? [/ QUOTE ] This is an interesting perspective. I might think that in a way, as strange as it might sound, the more blue cabs running at the time of the accident (up to some point), the more relevant is the witness. Cohen is probably exaggerating when arguing that the witness is all that matters, but maybe there's some way of assessing how relevant is the witness, as some function of the base rate? [/ QUOTE ] Well yea. There is a way of assessing it. It's called Bayes Theorum. PairTheBoard [/ QUOTE ] Bayes Theorem is NOT the entirety of what's going on here. The real question is whether the accuracy figure for the witness is "conditionally robust" or not. If it is, then the problem is internally self contradictory. If it's not, then you can potentially apply Bayes.
|
|
#18
02-09-2007, 05:43 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
Bayes Theorem is NOT the entirety of what's going on here. The real question is whether the accuracy figure for the witness is "conditionally robust" or not. If it is, then the problem is internally self contradictory. If it's not, then you can potentially apply Bayes. [/ QUOTE ] What do you mean by "conditionally robust"? I've never heard of that term before. PairTheBoard
|
|
#19
02-09-2007, 05:50 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
[ QUOTE ] [ QUOTE ] I think alThor's point is spot on. Suppose the prosecutor brings the witness who says he saw a blue cab and an expert who testifies that under controlled tests the witness has the 80% accuracy. At that point the jurors will weigh the evidence according to the 80% accuracy. But suppose the Defense then brings an expert who testifies that 0% of the cabs running that day were Blue. Clearly the Jurors will take that evidence into account and conclude the witness was mistaken. [/ QUOTE ] In this point of the trial it simply seems that the witness and the expert testifying about his accuracy are both completely irrelevant. The cab was green since it couldn't have been blue. [ QUOTE ] But now suppose instead that the Defense brings an expert who testifies that only 1% of the cabs running that day were Blue. Should the jurors not also take that evidence into consideration to conclude there is a high likelihood the Witness is mistaken? Surely even Cohen would agree they should. But what if it's 2%. Or 3%. Or 15%? At what point does Cohen argue that such evidence becomes irrelevant? [/ QUOTE ] This is an interesting perspective. I might think that in a way, as strange as it might sound, the more blue cabs running at the time of the accident (up to some point), the more relevant is the witness. Cohen is probably exaggerating when arguing that the witness is all that matters, but maybe there's some way of assessing how relevant is the witness, as some function of the base rate? [/ QUOTE ] Well yea. There is a way of assessing it. It's called Bayes Theorum. PairTheBoard [/ QUOTE ] I don't think you understood what I meant, or maybe I wasn't clear enough. Simple Bayes does not solve the problem I'm referring to, as in my opinion the relevancy of the witness is either 0% or 100%, while in fact you can come up with some (lets call it practical) solution where you attribute some different relevancy value to (a) and (c) in the original problem. This value could be a function of some specific relation between (a) and (c), for instance. Maybe this is vague, or maybe nonsensical, but IMHO the use of Bayes in this problem is not 100% reasonable either.
|
|
#20
02-09-2007, 05:55 PM
|
|||
|
|||
|
Re: The Cab Problem
[ QUOTE ]
[ QUOTE ] back to the original question, I just did some quick simulations in Excel to suggest the Bayesian reasoning is correct. Using 1000 "nights" with probability of 85% of it being a green car and 15% of being a blue car, I then had an observer pick the correct color 80% of the time and the wrong color 20% of the time. I then looked at (# of times blue car was claimed to be seen and it truly was the blue car)/(# of times blue car was claimed to be seen) and it was 40% which is right in line with 41%. [/ QUOTE ] I believe that this does not prove much, as it is simply going in a circle. Surely the Bayesian theorem solution to this problem is correct if you use the Bayesian theorem assumption to solve the problem. I don't see why you would need Excel to show that. [/ QUOTE ] Yes, it looks similar to Bayes rules but I'm just looking at the percent correct given that the witness said the car was blue and the 80/20 info(i.e. the probability that the cab was blue given all the information). I fail to see the existence of circular logic here.
|
 |
|
|
Linear Mode
