Back To The Shoe Argument

[David Sklansky]

Your chezlaw's cousin aren't you?

[BIG NIGE]

No.

[jason1990]

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Eyewitness testimony could never convict a person in your approach because they already knew what the defendant looks like. ??

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I have not put forth an approach. I am merely discussing David's mathematical model. But I like your post and I think it cuts straight to the heart of another subjective element in David's model: the definition of "evidence". To properly consider this, we must look at an equation. The revised probability of guilt in this example is

0.8/(0.8 + 0.2p),

where p is the probability the defendant has a size 13 shoe, given that he is innocent and given all the other "evidence".

As you pointed out, the absurd result in my hypothetical comes from defining "evidence" in the broadest sense as anything known to the jury. In this case, the evidence can include the fact that he has a size 13 shoe, so that p = 1 and the revised probability of guilt is the same as the old one.

David has tried to clarify things by defining "evidence" as only those things which "change the price". I assume he means only those things which affect the jury's assessment of the probability of guilt. However, this definition is too strict. For example, it may come to light in the trial that the police were biased and sought out a suspect who was of large stature, but that their bias was later found to be unjustified. This fact ought to increase p, even though it might not "change the price".

In the end, deciding what facts can and cannot be used to determine p is not at all a straightforward, objective procedure. The determination of p must be done subjectively and is the job of the jury. David originally said that p should be 0.01, since 1% of the general population has the murderer's shoe size. This is a natural assumption that one might make in a homework problem in an elementary probability class. But in real life, it is a gross oversimplification.

[jason1990]

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None of their Million alternate accused should have those attributes because they are not attributes of the murderer. They should have the known attributes of murderer

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Remember, the million consists of 800,000 guilty and 200,000 innocent. The 800,000 should have the attributes of the murderer. The 200,000 should have the attributes of the defendant, under the presumption he is innocent.

If there are known circumstances which make certain attributes of the defendant more or less likely, then these must be reflected in the composition of the 200,000 -- even if the circumstances have no bearing on the attributes of the murderer. One obvious example of such circumstances could be racial profiling.

[PairTheBoard]

[ QUOTE ]
it may come to light in the trial that the police were biased and sought out a suspect who was of large stature, but that their bias was later found to be unjustified. This fact ought to increase p, even though it might not "change the price".


[/ QUOTE ]

Why would it matter if it was unjustified? Justified or not, it would mean both guilty or innocent people arrested would all be of large stature. Thus changing the probability distribution of shoe sizes for the theoretical innocent group.

A couple of other points. First, suppose there is no Shoe Print Evidence. But suppose the jury knows this fact about unjustified Police Bias in the arrest. Shouldn't Sklansky have the Jury use this information to bias them toward acquittal? Isn't it more likely an innocent man was arrested because of Police Bias in the Arrest than if there had been no such Bias?

The Second Point involves this principle in the Race example. Aren't Police often biased and tend to arrest Black people more readily than other races? Doesn't this automatically affect p? And in the absence of the Witness, should the Jury use this belief to bias the defendant toward acquittal?



PairTheBoard

[PLOlover]

two points

1) No one has addressed that fact that under this model a black man can be convicted by an impartial (50-50 to start) jury given just the fact that a black man was seen at the crime. (actually if blacks are 10% of population then black men are only 5% so it is really like 97% certain or so under the model, not just 95%, which I think most people would agree is beyond a reasonable doubt.)

2) The whole statistical thing is very tricky for the following real life reason. (Well first just remember the duke rape case. The prosecutor withheld exculpatory evidence. Major big deal, he should be disbarred and jailed imo.)
Now what can happen is that the prosector can take a bunch of low level meaningless circumstantial evidence and only use that which applies to the defendant. More importantly, they can only look for such evidence, which would skirt the discovery process because they only have favorable evicdence for the prosecution.
Bascially cherry picking I guess.
Now with a competent defense attorney this can be fought, but without one the defendant can be totally railroaded and most people won't even notice, and the prosecutor can make the parlay claim and say it like 99% certain that all of these facts match.

[soon2bepro]

[ QUOTE ]
Do you agree or not that the rarer the shoe size, the greater the reason for the jury to convict?

[/ QUOTE ]

If the jury was conformed of rational beings (which it isn't) and laws were written in a rational manner, with the objective of maximizing "justice" (which they're not) the answer would be yes. As is, I really can't say.

[jason1990]

[ QUOTE ]
[ QUOTE ]
it may come to light in the trial that the police were biased and sought out a suspect who was of large stature, but that their bias was later found to be unjustified. This fact ought to increase p, even though it might not "change the price".


[/ QUOTE ]

Why would it matter if it was unjustified? Justified or not, it would mean both guilty or innocent people arrested would all be of large stature. Thus changing the probability distribution of shoe sizes for the theoretical innocent group.

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Correct. But if the bias were justified, then there must be evidence indicating that the murderer is more likely to be a large man. Since the defendant is also a large man, this might increase the likelihood of guilt (pre-footprint).

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A couple of other points. First, suppose there is no Shoe Print Evidence. But suppose the jury knows this fact about unjustified Police Bias in the arrest. Shouldn't Sklansky have the Jury use this information to bias them toward acquittal? Isn't it more likely an innocent man was arrested because of Police Bias in the Arrest than if there had been no such Bias?

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Not necessarily. If the murderer is a large man, then the police bias, justified or not, made them more likely to arrest the right man. If the murderer is a small man, then the police bias made them less likely to arrest the right man. If there is no evidence either way regarding the stature of the murderer, then we cannot say what effect the bias had.

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The Second Point involves this principle in the Race example. Aren't Police often biased and tend to arrest Black people more readily than other races? Doesn't this automatically affect p?

[/ QUOTE ]
If there is racial bias, then of course this affects p.

[ QUOTE ]
And in the absence of the Witness, should the Jury use this belief to bias the defendant toward acquittal?

[/ QUOTE ]
Not necessarily, for the same reasons as above. Here is a simplified scenario. Suppose the police are 100% biased and always arrest a black man. Also suppose for simplicity that in those cases where the murderer is black, the police always arrest the right man. Assume the jury knows all this. When a defendant comes before the court, what is the probability he is guilty? It is simply the probability that the murderer is black. If there is evidence that the murderer is black, then this is evidence in favor of conviction. If there is evidence that the murderer is not black, then this is evidence in favor of acquittal. Does the police bias affect the probability that the actual murderer was black? Evidently not. The police bias is independent of the race of the murderer. They are biased no matter what. So the police bias should not sway the jury in either direction when it comes to determining the likelihood that the actual murderer is black, and hence the likelihood of guilt in this particular case.

In general, police bias will affect the long term statistics of a sequence of trials by affecting what kinds of defendants are brought before the court. But its effect, if any, on a single trial depends entirely on the circumstances of that particular trial.

[molotom]

[ QUOTE ]
two points

1) No one has addressed that fact that under this model a black man can be convicted by an impartial (50-50 to start) jury given just the fact that a black man was seen at the crime. (actually if blacks are 10% of population then black men are only 5% so it is really like 97% certain or so under the model, not just 95%, which I think most people would agree is beyond a reasonable doubt.)


[/ QUOTE ]

I think part of this 'point' is an appeal to outrage w/ the emphasis on someone being unduly convicted because of their race. Nonetheless there is something to it. What I would point out is that the proportion of black people amongst the set of plausible suspects is not the same as that in the population at large. For instance, if the crime occured in a black district of a city in which the black population is poorer and more criminal, then the fact that this particular suspect is black is of less interest. Conversely, if the crime occured in a rural Japanese village, then the suspect being black would of course be a far stronger piece of evidence.

On the other hand, the distribution of shoe size amongst the suspects is probably closer to that of the population at large, because it's less correlated with likelihood of commiting a crime; although as someone mentioned this could be biased if the police were looking for a large man as the suspect.

Regardless, I think the spirit of the original question was that distribution of the attribute amongst the set of suspects was well-known, for example in the case where it is almost identical to that of the population at large (perhaps DNA evidence).

[jason1990]

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No one has addressed that fact that under this model a black man can be convicted by an impartial (50-50 to start) jury given just the fact that a black man was seen at the crime.

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In the footprint example, the situation is even more extreme. Imagine that without the footprint, the jury thought the defendant was 97% likely to be innocent. Then the footprint comes and it is size 14. That is about 3 standard deviations above the mean. The probability that a uniformly random person from the general population will have a size 14 shoe is about 0.00135. According to the model David proposes, the jury should now regard the probability of guilt as

0.03/(0.03 + 0.97(0.00135)) = 0.96.

In other words, the footprint alone is sufficient to convict the defendant. In fact, not only that, the footprint alone is sufficient to override whatever other evidence they have which convinced them he was 97% likely to be innocent. As you said, any random person with a size 14 shoe (and there are about 400,000 of them in the U.S.) could be convicted of this crime solely on the basis of his shoe size.

Arguably, this goes against the spirit of the court system. Since the model David proposes contains one or more subjective assumptions, this example could be used to argue that those assumptions are not necessarily consistent with the spirit of our legal system -- at least not in extreme examples such as this.

Let me say one more thing before I must unfortunately disappear for a while and do some actual work. When updating a probability assessment in light of new evidence, Bayes Theorem is the correct tool to use. When people try to use only their intuition, they frequently are led to numbers that are way off. It is important for people to learn how to use Bayes Theorem correctly so that they are not misled by their faulty intuition. David claimed that this is the point of the thread, so I want to finally reiterate that this is indeed the case.

However, this does not mean Bayes Theorem is a magical divining rod of ultimate truth. When David says "math tells us that [blank]" or "Bayes Theorem tells us that [blank]", this is not entirely true. What he should say is "Bayes Theorem tells us that my assumptions imply [blank]".

Bayesian analysis is always built upon assumptions. This is why the Bayesian interpretation of probability is also called the subjective interpretation. The output of a Bayesian analysis is only as good as the assumptions that are put into it.