Back To The Shoe Argument [Archive] - Two Plus Two Newer Archives

David Sklansky

05-24-2007, 06:45 PM

The first two responses are exhibiting some kind of math phobia or mental block. This is very disconcerting so I am going to try to show in precise detail why this guy deserves to be convicted.

Firstly lets say that the jury originally believed that there was a 20% chance he was innocent. What precisely do I mean by that? It means that the jury thinks that if they were presented with a million identical cases with the exact same evidence, 200,000 of the defendents would be innocent. First make sure you understand that definition.

OK, now assume that the shoe size fits one percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that shoe size. 200,000 are innocent. 2,000 have that shoe size. The other 198,000 are sent home. The trial now is for 802,000 defendents. All identical with the (adjusted) exact same evidence. Now the jury believes that one quarter of one percent of the defendents are innocent. (2000 out of 802,000).

Bringing it down to one defendent, the shoe evidence should change the juries probability of innocence from 20% to one fourth of one percent. The fact that juries may not think that way is neither here nor there.

mjkidd

05-24-2007, 06:59 PM

A small but pertinent hijack: Reasonable doubt is generally defined as:

"Any doubt which would make a reasonable person hesitate in the most important of his or her affairs."

Given this standard, what is the probability of innocence at which you would be comfortable convicting someone?

Taraz

05-24-2007, 07:00 PM

[ QUOTE ]
The first two responses are exhibiting some kind of math phobia or mental block. This is very disconcerting so I am going to try to show in precise detail why this guy deserves to be convicted.

Firstly lets say that the jury originally believed that there was a 20% chance he was innocent. What precisely do I mean by that? It means that the jury thinks that if they were presented with a million identical cases with the exact same evidence, 200,000 of the defendents would be innocent. First make sure you understand that definition.

OK, now assume that the shoe size fits one percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that shoe size. 200,000 are innocent. 2,000 have that shoe size. The other 198,000 are sent home. The trial now is for 802,000 defendents. All identical with the (adjusted) exact same evidence. Now the jury believes that one quarter of one percent of the defendents are innocent. (2000 out of 802,000).

Bringing it down to one defendent, the shoe evidence should change the juries probability of innocence from 20% to one fourth of one percent. The fact that juries may not think that way is neither here nor there.

Bringing it

[/ QUOTE ]

I would agree with you, but I don't think you can break down someone's probable guilt that clearly. For me it would depend on why I had reasonable doubt. Clearly the shoe-print evidence is huge, I think it would just depend on the particulars of the case.

For example, if the defendant admitted to being at the crime scene, the shoe-print is practically irrelevant. That's why I said it depends on so many other factors.

(Edit: Somehow I missed it when you said the following [ QUOTE ]
If it wasn't, its instant acquittal. But since it is the jury is now contemplating a conviction.

[/ QUOTE ] I guess I would agree with you in that case, but I still maintain that it would clearly depend on many factors.)

Piers

05-24-2007, 07:12 PM

The American legal system is beyond me, however the rarer the shoe size the more likely the accused is to be the murderer.

SNOWBALL

05-24-2007, 07:21 PM

[ QUOTE ]

For example, if the defendant admitted to being at the crime scene, the shoe-print is practically irrelevant. That's why I said it depends on so many other factors.

[/ QUOTE ]

did you read the OP?
DS said they know they footprint belonged to THE MURDERER.

Taraz

05-24-2007, 07:30 PM

[ QUOTE ]
[ QUOTE ]

For example, if the defendant admitted to being at the crime scene, the shoe-print is practically irrelevant. That's why I said it depends on so many other factors.

[/ QUOTE ]

did you read the OP?
DS said they know they footprint belonged to THE MURDERER.

[/ QUOTE ]

LOL, wow I'm blind.

bunny

05-24-2007, 08:07 PM

The guy is obviously more likely to be guilty the rarer the shoe size. How is this even debatable?

BIG NIGE

05-24-2007, 08:43 PM

[ QUOTE ]
I need to resurrect it with all this mathematical wrangling going on about evidence. I forget what people's original answers were.

Suppose a man is on trial for murder and the jury is on the verge of acquitting him in spite of their strong suspicion of guilt because the evidence leaves room for reasonable doubt. But at the last minute a footprint is uncovered at the murder scene. It is definitely the murderer's. And it is the same size as the defendent. If it wasn't, its instant acquittal. But since it is the jury is now contemplating a conviction.

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

[/ QUOTE ]

No. The jury should acquit no matter what unless the footprint is so large or small (or have any other kind of unique identification marking) that there's no other person in the area who could *reasonably* have it, thus removing the reasonable doubt. Of course, the whole premise of this question, or argument as you call it, is highly questionable, because it is hard to imagine a circumstance in which the footprint would "definitely" be the murderers, not to mention that it is HIGHLY improbable that an entire trial would go on and then all of a sudden a footprint is found after investigators have combed the area thoroughly multiple times, as is the case in murder investigations. At that point it would be almost as likely that while he was in jail or out on bail during the trial, he himself or someone else took the defendant's shoe and just planted it there.

jason1990

05-24-2007, 08:52 PM

[ QUOTE ]
OK, now assume that the shoe size fits one percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that shoe size.

[/ QUOTE ]
Okay.

[ QUOTE ]
200,000 are innocent. 2,000 have that shoe size.

[/ QUOTE ]
Not necessarily. If these 200,000 people were an unbiased random sample from the population at large, then yes, 2,000 would have that shoe size. But they are not a random sample. They are 200,000 people who are on trial for murder, who are faced with the same set of evidence against them (whatever that evidence may be), and who happen to be innocent. The proportion of them with the murderer's shoe size need not be 1%. It could be anything and would depend on a number of factors. For example, suppose the evidence indicates that the police suspected early on that the murderer was a large man. They therefore ruled out suspects that were of small to average stature. Hence, in those 200,000 instances where they picked the wrong man, their choice would have been biased toward larger men whose shoe size is larger than average.

Besides all this, though, there is another issue. There are not 1 million defendants. There is only 1. When you hypothesize 1 million identical defendants, you are simply building a probabilistic model. And you have built assumptions into that model. In this case, you assumed 2,000 out of the 200,000 innocents would have the murderer's shoe size. This may not be a good assumption. You could replace it with a different assumption if you like, but in the end, you will still only have a model of the situation which is based on subjective assumptions. It should be recognized as such, and not seen as some indisputable truth. The math may be indisputable, but the model is not.

LooseCaller

05-24-2007, 09:20 PM

score another one for realizing that pure probablistic models will always be flawed when imposed on reality.

PLOlover

05-24-2007, 09:24 PM

[ QUOTE ]

Bringing it down to one defendent, the shoe evidence should change the juries probability of innocence from 20% to one fourth of one percent. The fact that juries may not think that way is neither here nor there.

[/ QUOTE ]

I don't think most people think that way (.2)*(.01)=about1/4 of one percent

but rather people look at it like he's about 80% guilty, this new evidence points to 99% guilt, but the 80% guilty is an amalgamation of guilt approximations (66,70,90,95,50,45,88,90...)
so
when you throw in that 99 to the above data it won't sway it as much as you think.

I guess I'm saying that in your example you're giving the shoe data point as much weight as all the other evidence combined, 50% shoe, 50% rest.

---------------
Here's a counterexample that brings the 20% innocent down to 2.5%, which I would argue is "beyond a reasonable doubt".

instead of a shoeprint, at the last moment a witness comes forth who is certain of only one thing, the murderer was black. The defendant is also black. Let's say the percentage of blacks in the population is 10%.

OK, now assume that the RACE fits ten percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that RACE. 200,000 are innocent. 20,000 have that RACE. The other 180,000 are sent home. The trial now is for 820,000 defendents. All identical with the (adjusted) exact same evidence. Now the jury believes that 2.5% of the defendents are innocent. (20,000 out of 820,000).

David Sklansky

05-24-2007, 09:24 PM

I was going to point out that I am assuming that there is nothing about the case that would in any way point to the defendants shoe size. But I didn't want to make the lesson any more complicated. It appears you don't dispute that this technique is valid if the shoe size is independent. And you surely realize that most of those who would still argue against conviction are doing so not due to the technicalities you bring up, but rather because they have no clue how to apply the general math principles. But rather than fix that ignorance, they read your post and think they don't have to.

David Sklansky

05-24-2007, 09:30 PM

"No. The jury should acquit no matter what unless the footprint is so large or small (or have any other kind of unique identification marking) that there's no other person in the area who could *reasonably* have it, thus removing the reasonable doubt."

That's just wrong. And even the courts realize that, since they allow prosecutors to argue, when they have several pieces of independent but not by themselves totally damning evidence, that the PARLAY is damning. I think they may use different words though.

PLOlover

05-24-2007, 09:34 PM

let me put it another way. say the jury is 50-50, which is of course way reasonably doubtful. Let's put the threshhold at about 95% for conviction.

If at the last moment, a witness comes forward and says it was a black man, and the defendant is black, and blacks are about 10% of population, then presto, theyve reached the 95% threshold (ok approximately, but you see my point).

So a guy who was 50-50 gets convicted just because he's black?

PLOlover

05-24-2007, 09:46 PM

[ QUOTE ]
let me put it another way. say the jury is 50-50, which is of course way reasonably doubtful. Let's put the threshhold at about 95% for conviction.

If at the last moment, a witness comes forward and says it was a black man, and the defendant is black, and blacks are about 10% of population, then presto, theyve reached the 95% threshold (ok approximately, but you see my point).

So a guy who was 50-50 gets convicted just because he's black?

[/ QUOTE ]

So it seems to me that using this model, if you take a random black guy and accuse him of a crime that you have overwhemling evidence that a black guy committed, then the jury can have enough confidence to convict.

David Sklansky

05-24-2007, 09:47 PM

[ QUOTE ]
let me put it another way. say the jury is 50-50, which is of course way reasonably doubtful. Let's put the threshhold at about 95% for conviction.

If at the last moment, a witness comes forward and says it was a black man, and the defendant is black, and blacks are about 10% of population, then presto, theyve reached the 95% threshold (ok approximately, but you see my point).

So a guy who was 50-50 gets convicted just because he's black?

[/ QUOTE ]

If the blackness is a totally independent factor and 95% is the right threshold, then the answer should be yes. To see this, suppose only one in a million are black. You guys seem all to be forgetting that almost any piece of evidence is of this form. The make of car he was driving, his blood type. etc. etc.

This is not to say that I think all evidence should be admissable. If it will be misevaluated by dumb juries it might be better for them not to know it.

Leaky Eye

05-24-2007, 09:55 PM

As has been said, it totally depends on on the source of the current doubt. If the jury believes there is a 20% chance the police planted the evidence the shoe print is probably irrelevant. See the correct acquittal of OJ for this example.

If circumstantial evidence simply leaves 20% doubt, then the shoe print affects the outcome exactly as you describe.

PLOlover

05-24-2007, 09:58 PM

[ QUOTE ]
If the blackness is a totally independent factor and 95% is the right threshold, then the answer should be yes.

[/ QUOTE ]

Well I think this is proof that your model or whatever is simply not correct, since you can basically convict any random black guy of a crime committed by a black guy under your model.

absurdo reducto and all that

jason1990

05-24-2007, 10:50 PM

[ QUOTE ]
you surely realize that most of those who would still argue against conviction are doing so not due to the technicalities you bring up, but rather because they have no clue how to apply the general math principles. But rather than fix that ignorance, they read your post and think they don't have to.

[/ QUOTE ]
My goal today is not to teach people how to apply Bayes Theorem. That is a lesson for another day. Today's lesson is that your argument in this thread, and others like it in other threads, contain an intrinsic subjectivity. It would be far too easy for people to read your posts and think that this argument is based on ironclad, objective, perhaps even scientific principles, and that the conclusion is an indisputable consequence of mathematics. I want people to realize that there is something subjective hiding away inside this argument which defeats any claim to 100% objectivity.

[ QUOTE ]
I was going to point out that I am assuming that there is nothing about the case that would in any way point to the defendants shoe size. But I didn't want to make the lesson any more complicated. It appears you don't dispute that this technique is valid if the shoe size is independent.

[/ QUOTE ]
I dispute that this method is objective and scientific. As you suggest, let us assume that there is nothing about the case that would in any way point to the defendant's shoe size. You incorporate that fact into your probability model by assuming that, in the case the defendant is innocent, his shoe size is random and its distribution is the same as the distribution of the shoe size of a random person, uniformly selected from the population at large. This is the indifference principle. You are saying that "no information" = "uniform distribution". This is a subjective assumption. It is philosophical in nature. It is not science and it is not objective.

I agree that if we are forced to make some assumption about a probability distribution in the absence of any information, it is hard to think of anything better than a uniform distribution. But that does not make the uniform assumption any less subjective. In some cases, a better approach would be to make no assumption at all, and to simply acknowledge that formal probability theory and objective principles have nothing definitive to say about the matter at hand.

PairTheBoard

05-24-2007, 11:04 PM

[ QUOTE ]
Firstly lets say that the jury originally believed that there was a 20% chance he was innocent. What precisely do I mean by that? It means that the jury thinks that if they were presented with a million identical cases with the exact same evidence, 200,000 of the defendents would be innocent.

[/ QUOTE ]

First of all, I don't think people in general think in those terms. As mjkidd points out, these are the kind of terms they are given by the court,

[ QUOTE ]
Reaonable doubt -
"Any doubt which would make a reasonable person hesitate in the most important of his or her affairs."

[/ QUOTE ]

Those are terms which each juror will interpret subjectively. They are also charged with the instruction not to form an opinion about the case until they've seen all the evidence. They do not even begin Deliberations until then. During trial their job is to hear the evidence, not form opinions based on an incomplete knowledge of all the evidence. This is important so that their hearing of later evidence is not tainted by prejudice from forming prior opinions of guilt based on incomplete earlier evidence.

So it shouldn't matter whether the Shoe Size evidence comes early or late in the trial. And if it comes late in the trial it should not be subject to a premature opinion of guilt formed from incomplete evidence. After all the evidence is presented it should be weighed together.

It may not be the case that each bit of evidence acts independently of the others. The pieces of evidence may not simply add up like disjoint events of probability. There may very possibly be a complex interplay of the evidence, a kind of weave of evidence. The human capacities of intuition, personal experience, judgement, and reasoning working together to form a gestalt of conviction for guilt or innocence is not something easily modeled in terms of mathematical probabilities. If it were, we could produce good AI machines.

With all this in mind, the question becomes, is the Shoe Evidence relevant? On the surface, it seems to me that it is relevant circumstantial evidence. It looks similiar to fibre evidence where the fibre cannot be uniquely identified. The rarer the shoe size or fibre type, the more persuasive the circumstantial evidence. But in deliberations the jury must consider factors that impact this evidence. Could this be a case of mistaken identity? If the defendant only looks like the killer they are likely to have the same shoe size. Was the defendant framed? If so, the Shoe Evidence could be consistent with that theory. In fact, the rarity of the Shoe Size makes the planting of the imprint a good method for executing the frame on the defendant.

How many people wear that shoe size? Were there other possible suspects similiar in size to the defendant? Did the murder victim have friends, acquantances, or people she came in contact with recently who were the same size as the defendant? Are there known serial killers on the loose who have the same shoe size as the defendant? Have there been numerous unsolved cases where shoe imprints of this size have been left at the scene and for which the defendant has airtight alibis? Does this evidence actually have exonerative weight?

I suppose Skalnsky can try to redefine the question so as to purify it of all other considerations. But I think in doing so he misses my point which is that simple models of probability do not describe the process by which the jurors come to their decision. I give Sklansky the point that a Rare Shoe Size Imprint at the scene that matches the defendant's is relevant circumstantial evidence, just like fibre evidence. And the Rarer the Shoe Size the more persuasive the piece of evidence is. But if he wants something more precise than that he won't get it from me. I won't be pushed from that into statements about the evidence deciding guilt in a borderline case. In my opinion, such statements misrepresent the process.

PairTheBoard

Phil153

05-24-2007, 11:10 PM

Sklansky is owning this thread.

andyfox

05-24-2007, 11:14 PM

Not any random black guy. That he is black is part of the total evidence.

PLOlover

05-24-2007, 11:22 PM

[ QUOTE ]
Not any random black guy. That he is black is part of the total evidence.

[/ QUOTE ]

hope im being levelled here. in my simplist case the sole evidence is his blackness. so it really could be any random black guy.

brandofo

05-24-2007, 11:37 PM

[ QUOTE ]
Not any random black guy. That he is black is part of the total evidence.

[/ QUOTE ]
If the defendant is black, doesn't that remove any reasonable doubt?

PairTheBoard

05-24-2007, 11:40 PM

[ QUOTE ]
If the blackness is a totally independent factor

[/ QUOTE ]

How would you know whether it was or wasn't? If you don't see how race relates to the rigorous points jason1990 has made, you aren't paying attention.

PairTheBoard

jason1990

05-24-2007, 11:45 PM

luckyme

05-24-2007, 11:53 PM

[ QUOTE ]
But watch what happens when the second jury performs your thought experiment. ......

[/ QUOTE ]

It's not too late to delete is it?

luckyme

RED FACE

05-25-2007, 12:32 AM

I didn't read all the other replies but what if the defense counters this new evidence by proving that the victim had an enemy who had a friend with this same shoe size and no alibi?

If the case is built on circumstantial evidence then such a fact is significant.

jason1990

05-25-2007, 12:51 AM

[ QUOTE ]
[ QUOTE ]
But watch what happens when the second jury performs your thought experiment. ......

[/ QUOTE ]

It's not too late to delete is it?

[/ QUOTE ]
Are you suggesting I said something stupid and should delete my post to avoid embarrassment? If I have made a blunder, I do not mind admitting it later. But I am simply illustrating something that David has already acknowledged. Namely, if the pre-footprint evidence points to the shoe size of the defendant, then it will affect the outcome of the mathematical analysis. However, it is clear in the hypothetical that it should not affect the outcome of the actual analysis. This is because, at every point in time, both juries possess the same set of relevant evidence.

If the two situations produce the same actual conclusions, but the math gives two different conclusions, then the mathematical model must be either flawed or incomplete. This apparent contradiction requires an explanation and, I think, demonstrates that things are not quite as simple as they may seem.

PairTheBoard

05-25-2007, 01:21 AM

[ QUOTE ]
This apparent contradiction requires an explanation and, I think, demonstrates that things are not quite as simple as they may seem.

[/ QUOTE ]

It especially requires explanation in consideration of the Race example. It's a hypothetical that the shoe size might be known prior to the late shoe imprint evidence. But the race of the defendent is clearly apparent to the Jury prior to the late witness evidence.

I could explain it but I'd like to see Sklansky do it. Maybe it will force him to take a second look at his hidden assumptions.

PairTheBoard

luckyme

05-25-2007, 02:07 AM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
But watch what happens when the second jury performs your thought experiment. ......

[/ QUOTE ]

It's not too late to delete is it?

[/ QUOTE ]
Are you suggesting I said something stupid and should delete my post to avoid embarrassment? If I have made a blunder, I do not mind admitting it later. But I am simply illustrating something that David has already acknowledged. Namely, if the pre-footprint evidence points to the shoe size of the defendant, then it will affect the outcome of the mathematical analysis. However, it is clear in the hypothetical that it should not affect the outcome of the actual analysis. This is because, at every point in time, both juries possess the same set of relevant evidence.

If the two situations produce the same actual conclusions, but the math gives two different conclusions, then the mathematical model must be either flawed or incomplete. This apparent contradiction requires an explanation and, I think, demonstrates that things are not quite as simple as they may seem.

[/ QUOTE ]

Essentially you're swinging between attributes of this specific defendant and attributes of the actual murderer ( and therefore potential murderers)and morphing them. Their original Million potential murderers shouldn't have size 13 shoes just because this specific defendant does.

In your second trial the jury noticed that the man had red hair, tattoo on nose and his left eye missing but there was no testimony that the murderer had those attributes. Why would their Million alternates have those features. They've reached the 80% convinced of quilt stage on other evidence. 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 and could well be the same Million in DS's first case. Essentially, the Million alternates are blankfaced and shoeless until there is evidence that the murderer had X ( slightly simplified).

Late evidence of a video show a man looking like the defendant comes in. Are you sure their 80% confidence in quilt doesn't change?

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

luckyme

David Sklansky

05-25-2007, 02:16 AM

"So it shouldn't matter whether the Shoe Size evidence comes early or late in the trial. And if it comes late in the trial it should not be subject to a premature opinion of guilt formed from incomplete evidence. After all the evidence is presented it should be weighed together"

It doesn't matter whether the shoe size evidence comes early or late. The end result is the same. If it was the absolutely first piece of evidence he would still be a fifty million to one UNDERDOG to be guilty at that moment (disregarding the fact he was arrested etc.). Then the rest of the evidence would bring him down up to 99.7%

The fact of the matter is that the mathematical technique that makes use of ratios of probabilities ( I won't use the B word) is very useful. And most people don't have a clue how to do this and therefore often make wildly wrong probability judgements. It is true that the technique makes assumptions but they are usually reasonable. And the errors made because of imperfect assumptions pale in comparison to the errors made by people who have no idea how to use the technique.

David Sklansky

05-25-2007, 02:27 AM

Bingo. I take back everything I ever said about you.

Putting it slightly differently. First remember that I define evidence as "anything that changes the price". Now remember that I postulated a trial of 1,000,000 people where EVIDENCE is identical. Not everything about them. Including at this point the shoe size. Thus the million defendents would not all have to have the same size shoe as the defendent.

What's the chances that luckyme would find a flaw in a professional mathmetician's argument about Baye's Theorem? This tells me there may indeed be a God.

BIG NIGE

05-25-2007, 02:42 AM

[ QUOTE ]
"No. The jury should acquit no matter what unless the footprint is so large or small (or have any other kind of unique identification marking) that there's no other person in the area who could *reasonably* have it, thus removing the reasonable doubt."

That's just wrong. And even the courts realize that, since they allow prosecutors to argue, when they have several pieces of independent but not by themselves totally damning evidence, that the PARLAY is damning. I think they may use different words though.

[/ QUOTE ]

Okay your point is taken, but I am still skeptical of your premise that the footprint was just found after a likely lengthy period of time between the crime scene investigation and the end of the trial, and also that it was "definitely" the murderer's. How does something as obvious to look for as a footprint just go unnoticed? And even if it really did go unnoticed, how could one be sure that it was even there all along, and not a new footprint? If I was investigating and I saw a footprint at a crime scene weeks or months after the investigation, my first guess would be that someone probably steped there recently. I just don't see how a footprint could be "overlooked". Now sure it may be that it was barely detectable and you needed some advanced science to identify it and map it out, but the trial would never have reached its near conclusion without the footprint, if the prosecution thought it such a damning piece of evidence. The only thing I can think of is if this hypothetical scenario you propose is an allegory for a broader concept or idea, but if so then I don't know what that is. Although I didn't read the whole thread only the OP.

David Sklansky

05-25-2007, 02:51 AM

Your chezlaw's cousin aren't you?

BIG NIGE

05-25-2007, 02:53 AM

No.

jason1990

05-25-2007, 03:59 AM

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

[/ QUOTE ]
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

05-25-2007, 04:16 AM

[ QUOTE ]
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

[/ QUOTE ]
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

05-25-2007, 04:45 AM

[ 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

05-25-2007, 11:53 AM

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

05-25-2007, 11:58 AM

[ 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

05-25-2007, 12:10 PM

[ 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.

[/ QUOTE ]
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).

[ QUOTE ]
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?

[/ QUOTE ]
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.

[ QUOTE ]
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

05-25-2007, 12:25 PM

[ 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

05-25-2007, 01:01 PM

[ QUOTE ]
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.

[/ QUOTE ]
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.

PLOlover

05-25-2007, 01:54 PM

[ QUOTE ]
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.

[/ QUOTE ]

yes, this was my main point. DS's model breaks down in these situations so it can't be correct imo.

relativity_x

05-25-2007, 02:19 PM

Suppose the murder occurred in a commonly traveled location by the alleged killer. If he/she admits to being in the area, then of course his/her footprint will be there. This does not prove that the alleged killer is guilty. On the other hand, the evidence does locate the alleged around the scene, but this is no reason to convict the killer.

The footprint would have to be traced back to a particular time. If this footprint could be traced back to the date/time and if the time coincided with the death of the victim, then yes, I believe it would be sufficient evidence to convict the alleged murder.

jason1990

05-25-2007, 02:36 PM

[ QUOTE ]
Bingo. I take back everything I ever said about you.

Putting it slightly differently. First remember that I define evidence as "anything that changes the price". Now remember that I postulated a trial of 1,000,000 people where EVIDENCE is identical. Not everything about them. Including at this point the shoe size. Thus the million defendents would not all have to have the same size shoe as the defendent.

What's the chances that luckyme would find a flaw in a professional mathmetician's argument about Baye's Theorem? This tells me there may indeed be a God.

[/ QUOTE ]
Do you consider the shoe size of the murderer as "evidence"? In isolation, neither the defendant's nor the murderer's shoe size can "change the price". Only when both are known do we get relevant information. So if the defendant's shoe size is not evidence, then the murderer's cannot be either. Now consider your 800,000 hypothetical guilty defendants. What do we know about them? We know they are guilty and we know the evidence against them. Neither the defendant's nor the murderer's shoe size are considered evidence. So the shoe sizes of the 800,000, according to the model, are distributed like the population at large -- just like the shoe sizes of the 200,000.

We now factor in the new evidence, which is that the defendant and the murderer both have shoe size X, which is the shoe size of only 1% of the population. We can therefore ignore any of the hypothetical trials in which this evidence is not met. This means we keep 2,000 of the innocent defendants and 8,000 of the guilty ones. Again the assessment has not changed. It is still 80% guilty. Clearly this is the wrong conclusion, so we must still not have the right definition of "evidence". We must find a definition of "evidence" which justifies incorporating the shoe size of the murderer into the 800,000 but ignoring the shoe size of the defendant in the 200,000.

It is possible to modify the model to make this work, but the modification I have in mind does not involve redefining the word "evidence".

The point here is simply that the model contains many subjective assumptions and is easily attacked by reasonable objections -- objections which force the model to be clarified and redefined. You said in another thread,

[ QUOTE ]
Those who do disagree are frankly too dumb to discuss anything further with.

[/ QUOTE ]
Some people may raise objections which are ignorant and just plain wrong. But not all objections are of this variety. I agree that your example, your model, and your argument are absolutely superb when it comes to demonstrating the mathematical techniques involved in applying Bayes Theorem. But it does not demonstrate a universal, practical, and objective applicability of the method in the real world. Demonstrating this appears to be at least part of your goal here.

DMACM

05-25-2007, 03:02 PM

I dont see why theres any question. Of course the vague concept of "reasonable doubt" means we are drawing the line somewhere. We rationalize it is worse to let 10 guilty men go free then wrongly convict one. We can never have certainty.

A rarer size brings us closer to the line past which we convict.

PairTheBoard

05-25-2007, 04:27 PM

[ QUOTE ]
Sklansky is owning this thread.

[/ QUOTE ]

Still think Sklansky is owning this thread phil? With his limited knowledge of mathematical probability Sklansky has been able to Bully a lot of people around here on issues like this. That's because most people know even less than he does about mathematical probability. Some of them really are ignorant. But many actually have the correct intuition that Sklansky sometimes gets offbase on these things. They just don't have the expertise to argue with him about it.

When he goes up against someone who really is an expert in mathematical probability and cares to take the time to deal with him he is seen for what he is. A Bully carrying an "I'm smarter than you" Whip.

PairTheBoard

David Sklansky

05-25-2007, 04:31 PM

This whole technical debate is nonsense. It would go away instantly if I was careful to use the right words and the right scenario. And if I did the answer would be just as astonishing to people who don't understand Baye's Theorem.

PairTheBoard

05-25-2007, 05:16 PM

[ QUOTE ]
This whole technical debate is nonsense. It would go away instantly if I was careful to use the right words and the right scenario.

[/ QUOTE ]

What. We're supposed to take your word for that? Why? Is the debate really just "technical"? Bayes Theorem is "technical" to begin with. You are the one presenting a "technical" issue. But you're presenting it in a nontechnical setting. You object to observations about your technical tool because they are technical and then object further when we bring in issues related to the nontechnical setting because they are real life issues unrelated to the technical validity of your tool. You then declare things like, "everybody knows what I'm talking about", "anyone who doesn't get it is stupid", and "If I did it right you would see that I'm right". You are a piece of work David.

PairTheBoard

PLOlover

05-25-2007, 05:47 PM

[ QUOTE ]
What. We're supposed to take your word for that? Why? Is the debate really just "technical"? Bayes Theorem is "technical" to begin with. You are the one presenting a "technical" issue. But you're presenting it in a nontechnical setting. You object to observations about your technical tool because they are technical and then object further when we bring in issues related to the nontechnical setting because they are real life issues unrelated to the technical validity of your tool. You then declare things like, "everybody knows what I'm talking about", "anyone who doesn't get it is stupid", and "If I did it right you would see that I'm right". You are a piece of work David.

PairTheBoard

[/ QUOTE ]

I agree with DS, it's pretty easy to figure out what he's saying/ meant to say.

PairTheBoard

05-25-2007, 05:57 PM

[ QUOTE ]
I agree with DS, it's pretty easy to figure out what he's saying/ meant to say.

[/ QUOTE ]

Really? Then maybe you could tell us exactly what it is.

PairTheBoard

andyfox

05-25-2007, 06:08 PM

David has spelled it out exactly:

"lets say that the jury originally believed that there was a 20% chance he was innocent. What precisely do I mean by that? It means that the jury thinks that if they were presented with a million identical cases with the exact same evidence, 200,000 of the defendents would be innocent. First make sure you understand that definition.

"OK, now assume that the shoe size fits one percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that shoe size. 200,000 are innocent. 2,000 have that shoe size. The other 198,000 are sent home. The trial now is for 802,000 defendents. All identical with the (adjusted) exact same evidence. Now the jury believes that one quarter of one percent of the defendents are innocent. (2000 out of 802,000).

"Bringing it down to one defendent, the shoe evidence should change the juries probability of innocence from 20% to one fourth of one percent. The fact that juries may not think that way is neither here nor there."

Knowing the math makes one much more likely to make the correct decision. How much clearer can David be?

PairTheBoard

05-25-2007, 07:07 PM

[ QUOTE ]
David has spelled it out exactly:

"lets say that the jury originally believed that there was a 20% chance he was innocent. What precisely do I mean by that? It means that the jury thinks that if they were presented with a million identical cases with the exact same evidence, 200,000 of the defendents would be innocent. First make sure you understand that definition.

"OK, now assume that the shoe size fits one percent of the population. Now lets go back to the million identical defendents. 800,000 are guilty. All 800,000 have that shoe size. 200,000 are innocent. 2,000 have that shoe size. The other 198,000 are sent home. The trial now is for 802,000 defendents. All identical with the (adjusted) exact same evidence. Now the jury believes that one quarter of one percent of the defendents are innocent. (2000 out of 802,000).

"Bringing it down to one defendent, the shoe evidence should change the juries probability of innocence from 20% to one fourth of one percent. The fact that juries may not think that way is neither here nor there."

Knowing the math makes one much more likely to make the correct decision. How much clearer can David be?

[/ QUOTE ]

I suggest you read the objections to this approach made by people who "know the math" much much better than David does. If it was that "clear" we would not be making the objections.

PairTheBoard

PLOlover

05-25-2007, 07:27 PM

[ QUOTE ]
Quote:
I agree with DS, it's pretty easy to figure out what he's saying/ meant to say.

Really? Then maybe you could tell us exactly what it is.

PairTheBoard

[/ QUOTE ]

he asserts that circumstantial evidence is a lot stronger than most people think it is and then gives us his argument in statistical form.

PairTheBoard

05-25-2007, 08:09 PM

[ QUOTE ]
[ QUOTE ]
Quote:
I agree with DS, it's pretty easy to figure out what he's saying/ meant to say.

Really? Then maybe you could tell us exactly what it is.

PairTheBoard

[/ QUOTE ]

he asserts that circumstantial evidence is a lot stronger than most people think it is and then gives us his argument in statistical form.

[/ QUOTE ]

There could be cases where it's weaker than people think based on this approach as well. I don't think anyone here is contesting the general idea that everyone should understand Conditional Probability as a basic tool for turning data into information. But David goes farther than that and insists on imposing on us an artificial idealized probability model for judging the totality of evidence in a criminal case. It's very iffy whether such a thing is even feasible and even more iffy to propose that it would manufacture better results than human beings weighing the totality of evidence as they normally do.

The assertion that they will be better able to process data into information if they know Conditional Probability is like saying a mechanic can do a better job if he knows basic physics. That doesn't mean you would want your car fixed by somebody who can't take his mind off of nuclear physics.

PairTheBoard

luckyme

05-25-2007, 08:28 PM

[ QUOTE ]
The assertion that they will be better able to process data into information if they know Conditional Probability is like saying a mechanic can do a better job if he knows basic physics. That doesn't mean you would want your car fixed by somebody who can't take his mind off of nuclear physics.

PairTheBoard

[/ QUOTE ]

Or giRls. or FooTBall. or renT coming due.

Sure, hang around some trades people for a while, you would appreciate thoSe who undersTand 'basic physIcs'.

Yes, I would like my juries to understand simple conditional probability ( rather than the math prof's that screwed up Monte Hall) and some chemistry and a touch of physics and anything else that helps them think through the evidence better....

why would I shiver at that thought? I should want my mechanic to be Newtonless just to ensure he doesn't try and solve relativity problems when listening to my converter? I'd rather take away his Ipod and spray paint his Playboy cut out.

luckyme

PairTheBoard

05-25-2007, 08:49 PM

[ QUOTE ]
[ QUOTE ]
The assertion that they will be better able to process data into information if they know Conditional Probability is like saying a mechanic can do a better job if he knows basic physics. That doesn't mean you would want your car fixed by somebody who can't take his mind off of nuclear physics.

PairTheBoard

[/ QUOTE ]

Or giRls. or FooTBall. or renT coming due.

Sure, hang around some trades people for a while, you would appreciate thoSe who undersTand 'basic physIcs'.

Yes, I would like my juries to understand simple conditional probability ( rather than the math prof's that screwed up Monte Hall) and some chemistry and a touch of physics and anything else that helps them think through the evidence better....

why would I shiver at that thought? I should want my mechanic to be Newtonless just to ensure he doesn't try and solve relativity problems when listening to my converter? I'd rather take away his Ipod and spray paint his Playboy cut out.

luckyme

[/ QUOTE ]

You misunderstood me. Did you read this part of my post...

[ QUOTE ]
I don't think anyone here is contesting the general idea that everyone should understand Conditional Probability as a basic tool for turning data into information.

[/ QUOTE ]

Of oourse you want your mechanic to understand basic physics.

Maybe this is a better analogy. It's probably helpful for a Gourmet Chef to understand some basic chemistry. But do you want him preparing your meal according to some elaborate calculations of chemical reactions or according to his best judgment as to how things are going to taste?

Do you want a nuclear physicist working on your car or a mechanic who knows some basic physics?

PairTheBoard

luckyme

05-25-2007, 08:56 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
The assertion that they will be better able to process data into information if they know Conditional Probability is like saying a mechanic can do a better job if he knows basic physics. That doesn't mean you would want your car fixed by somebody who can't take his mind off of nuclear physics.

PairTheBoard

[/ QUOTE ]

Or giRls. or FooTBall. or renT coming due.

Sure, hang around some trades people for a while, you would appreciate thoSe who undersTand 'basic physIcs'.

Yes, I would like my juries to understand simple conditional probability ( rather than the math prof's that screwed up Monte Hall) and some chemistry and a touch of physics and anything else that helps them think through the evidence better....

why would I shiver at that thought? I should want my mechanic to be Newtonless just to ensure he doesn't try and solve relativity problems when listening to my converter? I'd rather take away his Ipod and spray paint his Playboy cut out.

luckyme

[/ QUOTE ]

You misunderstood me. Did you read this part of my post...

[ QUOTE ]
I don't think anyone here is contesting the general idea that everyone should understand Conditional Probability as a basic tool for turning data into information.

[/ QUOTE ]

Of oourse you want your mechanic to understand basic physics.

Maybe this is a better analogy. It's probably helpful for a Gourmet Chef to understand some basic chemistry. But do you want him preparing your meal according to some elaborate calculations of chemical reactions or according to his best judgment as to how things are going to taste?

Do you want a nuclear physicist working on your car or a mechanic who knows some basic physics?

PairTheBoard

[/ QUOTE ]

No. I understood your intitial point clearly and agreed with it. I disagreed with your slippery slope argument about it. Still do.

luckyme

David Sklansky

05-25-2007, 09:31 PM

PairTheBoard,

There are people coming to my defense who have a long history of disagreeing with me about similar ideas. I therefore feel no need to further clarify what I am getting at just to satisfy you or the mathmetician who has made some technical objections. You are embracing those objections because it satisfies your agenda which is different from jasons.

Your actual agenda stems from the fact that you are a nice guy. Probably nicer than everyone here. And it pains you to see hurt and suffering in anyone who doesn't deserve it. Even those who don't believe in Jesus. And even those who have difficulty with mathematical concepts. So you fight tooth and nail on their behalf. Perhaps subconsciously.

m_the0ry

05-25-2007, 09:36 PM

Legalism tries to look like math but no matter how hard it tries it will always have a gray area where everything is a semantic argument. It's defined by the definition of the language it uses.

PairTheBoard

05-25-2007, 10:18 PM

[ QUOTE ]
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.

[/ QUOTE ]

I don't think this observation has been given enough consideration. It's a major legitimate objection to giving too much weight to circumstantial evidence. It's basically the "Bible Code" phenomenon. As Siegmund points out in this thread by Pokerlogist on the Probability Forum,

[ QUOTE ]
an individual example, after the fact, looks peculiar, but something similar virtually certain to have occurred somewhere sometime.

[/ QUOTE ]

Probability Forum Thread - Coincidence or Evidence? (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=10524354&an=0&page=0#Pos t10524354)

We never know how much the Bible-Code effect is coming into play with this kind of circumstantial evidence. This is another reason why human judgement should sometimes overrule the kind of naked application of Conditional Probability that Sklansky advocates in the OP. It may be there were countless pieces of circumstantial evidence that could not be linked to the defendant and the only one the prosecution presented at trial was the one that could.

PairTheBoard

PairTheBoard

05-25-2007, 10:26 PM

[ QUOTE ]
PairTheBoard,

There are people coming to my defense who have a long history of disagreeing with me about similar ideas. I therefore feel no need to further clarify what I am getting at just to satisfy you or the mathmetician who has made some technical objections. You are embracing those objections because it satisfies your agenda which is different from jasons.

Your actual agenda stems from the fact that you are a nice guy. Probably nicer than everyone here. And it pains you to see hurt and suffering in anyone who doesn't deserve it. Even those who don't believe in Jesus. And even those who have difficulty with mathematical concepts. So you fight tooth and nail on their behalf. Perhaps subconsciously.

[/ QUOTE ]

I don't know why you think you can read my mind or know what my "agenda" is. As far as your appeal to your supporters on this thread, I see a lot of people saying things like this,

[ QUOTE ]
a Rare Shoe Size Imprint at the scene that matches the defendant's is relevant circumstantial evidence, just like fibre evidence. And the Rarer the Shoe Size the more persuasive the piece of evidence is.

[/ QUOTE ]

In fact, I said that myself. But I see a lot of objections to your Idealized Model for computing guilt or innocence.

PairTheBoard

PLOlover

05-26-2007, 09:25 AM

[ QUOTE ]
Quote:

Quote:
Quote:
I agree with DS, it's pretty easy to figure out what he's saying/ meant to say.

Really? Then maybe you could tell us exactly what it is.

PairTheBoard

he asserts that circumstantial evidence is a lot stronger than most people think it is and then gives us his argument in statistical form.

There could be cases where it's weaker than people think based on this approach as well. I don't think anyone here is contesting the general idea that everyone should understand Conditional Probability as a basic tool for turning data into information. But David goes farther than that and insists on imposing on us an artificial idealized probability model for judging the totality of evidence in a criminal case. It's very iffy whether such a thing is even feasible and even more iffy to propose that it would manufacture better results than human beings weighing the totality of evidence as they normally do.

The assertion that they will be better able to process data into information if they know Conditional Probability is like saying a mechanic can do a better job if he knows basic physics. That doesn't mean you would want your car fixed by somebody who can't take his mind off of nuclear physics.

PairTheBoard

[/ QUOTE ]

Well what I mean is that I understand his argument. Obviously I don't agree with it and I have stated my reasons why, basically
a) his argument can lead to reducto absurdo type of conclusions
b) cherry picking which you touched upon

wtfsvi

05-26-2007, 12:51 PM

If I'm going to committ a murder and leave foot prints in the blood, I will wear shoes that don't fit.

edit: If your scenario was not so flawed because of things like that and a lot of other things mentioned in this thread, nobody would disagree with you.

[ QUOTE ]
And if I did the answer would be just as astonishing to people who don't understand Baye's Theorem.

[/ QUOTE ] This doesn't seem true to me. I think very few (and most f them would be retarded people whom I assume would not be on a jury) will be surprised to know that rarer shoe size = more likely it is the defendant's shoe.

jason1990

05-26-2007, 01:09 PM

[ QUOTE ]
This whole technical debate is nonsense. It would go away instantly if I was careful to use the right words and the right scenario.

[/ QUOTE ]
The last time I remember you saying something like this, you said that epsilon-delta proofs are nonsense. It should come as no surprise that I disagree with you in both cases. In any event, you posted that you feel no need to further clarify what you are getting at. So I assume you will not be posting a correction to your model which "uses the right words." This surprises me, since I thought you would be interested in teaching the readership of this forum some elementary probability. But if you will not do it, then I will.

It is not enough to consider 800,000 guilty defendants and 200,000 innocent ones. When the defendant is innocent, that means there is a murderer on the loose somewhere. So we must pair each of the 200,000 innocent defendants with a hypothetical murderer. We will declare that all 1.2 million of these people have shoe sizes that are distributed like the general population.

When we learn that the defendant and the murderer both have shoe size X, we can begin the process of eliminating some of those hypotheticals. We discard all of the guilty defendants who do not have shoe size X. This leaves 8,000 guilty defendants. Among the innocent defendants, we keep only those who (a) have shoe size X, and (b) are matched up to a murderer with shoe size X. The number of such cases is 1% of 1% of 200,000, which is 20. In the end, the ratio of innocent to guilty defendants is 20 to 8,000, making the probability of innocence one quarter of one percent.

When we build the model properly in this way, there is no need to carefully define what is "evidence". For example, we can freely allow the defendant's shoe size to be part of the original evidence. In that case, all 1 million defendants have shoe size X, but the 200,000 murderers have random shoe sizes. When we learn that the murderer also has shoe size X, we discard all innocent defendants who are matched up to a murderer with a different shoe size. This leaves 2,000 innocent defendants for a resulting ratio of 2,000 to 800,000, giving the same probability.

Similarly, we could allow only the murderer's shoe size to be in the original evidence. In this case, all 800,000 guilty defendants and all 200,000 murderers have shoe size X, but the 200,000 innocent defendants have random shoe sizes. When we learn that the defendant also has shoe size X, we again get a 2,000 to 800,000 ratio.

This is the right way to avoid the contradictions that arise in this problem, rather than trying to find some artificial definition of "evidence". The key is to randomize not only the defendant, but also the murderer, and to recognize that in the case of innocence these are two different people. My apologies to anyone who finds this too technical. But I think if you take the time to carefully consider this example, you will learn something about how to do a proper probabilistic analysis.

A side remark: for future reference, let me point out that this post has nothing to do with my "agenda".

jason1990

05-26-2007, 01:09 PM

[ QUOTE ]
I therefore feel no need to further clarify what I am getting at just to satisfy you or the mathmetician who has made some technical objections. You are embracing those objections because it satisfies your agenda which is different from jasons.

[/ QUOTE ]
You mention that I have an agenda, but you have not described it or acknowledged it anywhere, except to say here that it is a technical objection. It is not a technical objection. It is not even a mathematical objection.

Strip away all the distracting details and boil this down to its essence: You are on a jury. The trial has not even started. You know nothing. You have absolutely no information. Except one thing: there is a murderer. Nothing else. At all.

What is the probability the murderer's shoe size is 13?

The only objective answer is: "I don't know." The Sklansky answer is: 1%, since 1% of the general population has shoe size 13.

The Sklansky answer is an opinion. It may be a common opinion. Many people may share this opinion. But it is still an opinion. It is a subjective belief. An assumption. My primary agenda in this thread was simply to make people aware of this.

David Sklansky

05-26-2007, 04:29 PM

When I said your PTB's agenda was different than yours I was not really implying that you had any agenda. Other than to make some technical points.

Do you agree with me that PTB is seizing on your technical objections as a way to justify assertions that far go beyond your points? Sort of like how religious people think erroneously that quantum theory backs up their ideas.

Do you agree that the vast vast majority of the people who are claiming that independent evidence should not have a great effect on probability of guilt even when the evidence is rare are saying that out of confusion rather than because I am assuming some indifference principle that they would have not trouble assuming themselves?

PS What epsilon delta are you talking about?

TomCowley

05-26-2007, 05:59 PM

Jason, all you did was jump through completely unnecessary hoops, divide all sklansky's numbers by 100 for no necessary reason, and then come up with the same answer.

DUCY?

Since you obviously don't, I'll spell it out for you.

Imagine a universe of 1 million people, 800k actual murderers, 200k actual innocents, which is the distribution specified in the problem.

Select a person at random. You have a 20% chance of picking an innocent.

Now throw out everybody without the proper shoe size. All the (800k) actual murderers have it. 1%*200k=2k innocents have it. Now the pool is 800k murderers and 2k innocents.

Pick one at random. 2k/802k = ~0.25% of getting an innocent.

Or in short terms:
800k murderers, 200k innocents.

Select all cases where the murderer has shoe size 13 (all of them, since it's assumed in the problem).

Then select all cases where the defendant has shoe size 13(800k:2k).

That's Sklansky.

Your argument is to imagine a universe of 1 million people that includes murderers without shoe size 13 (did you even read the problem?)

Now, in your 800k:200k universe, when you select cases where the murderer has shoe size 13, you get 8k murderers and 2k innocents. (8k actual murderers do, and 2k innocents are "paired" with murderers who do)

Now when you select for people with shoe size 13 given that murderer has shoe size 13, you get 20 innocents and the same 8k murderers.

Your universe, since you can't read the problem, is composed of 990k people =99% people WHO HAVE NOTHING TO DO WITH THE QUESTION SINCE THE MURDERER IN THEIR CASE ISN'T SIZE 13. There's absolutely no reason whatsoever to consider these people in the context of sklansky's question. It's just jumping through an additional hoop to throw them out (divide both numbers by 100) to get the EXACT SAME answer becuase sklansky's method, for the problem he described, is correct.

Shine

05-26-2007, 06:03 PM

DS,

Please respond to the argument that any black person can be convicted.

TomCowley

05-26-2007, 06:11 PM

You don't need Sklansky.

If the evidence makes it 80/20 independent of race (and assuming no correlation of race and crime), and then you find out the criminal and defendant are both black, it becomes 80:2 which is pretty strong. It's strong because the black guy was ALREADY 80% likely to be guilty.

If you take a random black, with no evidence at all, out of a population of 1 million (10% black), his "before" guilt is 1/1 million (somebody did it). Knowing a black did it simply makes his guilt 1/100000, which is the exact likelihood of picking that black out of the population of blacks, which is what you should expect hsi guilt to be with no other evidence. There's nothing fishy going on.

PairTheBoard

05-26-2007, 06:15 PM

There's no question that those calculations can be done on that Model. The problem is whether that Model actually fits reality. Consider this Model. There are 1 million simliar cases which the police investigate. The evidence they bring prior to the Shoe Size matches 800,000 guilty defendants and 200,000 innocent defendants. Now, for those 1 million different defendants the police look for additional circumstances that matches them to the crime. They find that additional circumstance for All 1 million of those defendants. The Trial we are in just happens to be the one where the matching circumstance is the Shoe Size. Now do your calculations.

PairTheBoard

TomCowley

05-26-2007, 06:36 PM

Any "perfectly black or white" evidence where the guilty/innocent must match/not match, such as shoe size, isn't subject to cherry picking. If there were ANY evidence of this type that didn't add up, it's an instant acquittal. That's what pre-trial disclosure is for. If you want to argue about likelihoods of corrupt prosecutors coupled with bad representation, go somewhere else. That isn't what this question is about.

PairTheBoard

05-26-2007, 07:16 PM

[ QUOTE ]
Any "perfectly black or white" evidence where the guilty/innocent must match/not match, such as shoe size, isn't subject to cherry picking. If there were ANY evidence of this type that didn't add up, it's an instant acquittal. That's what pre-trial disclosure is for. If you want to argue about likelihoods of corrupt prosecutors coupled with bad representation, go somewhere else. That isn't what this question is about.

[/ QUOTE ]

The question is about whether Skalnsky's model is accurate and whether the evidence really is independent. It's not just the evidence brought by the Prosecutors, it's the evidence ignored or collected by the police. I'm not saying the Shoe Size is irrelevant. Only that a judgment needs to be made that goes beyond the straight math model.

PairTheBoard

TomCowley

05-26-2007, 07:23 PM

You're not even making any sense. Every copy of the case has the same evidence, otherwise they wouldn't be copies of the case, and wouldn't be a meaningful way to describe an 80% chance of guilt in a SINGLE case. How can you POSSIBLY be considering cases with DIFFERENT evidence when there is exactly ONE defendant with ONE set of evidence?

And if your argument is only "Police/prosecution intentionally ignore/withhold exculpatory black and white evidence, and the model doens't account for this," STFU because you're a complete moron incapable of basic abstract thought.

PairTheBoard

05-26-2007, 07:44 PM

[ QUOTE ]
You're not even making any sense. Every copy of the case has the same evidence, otherwise they wouldn't be copies of the case, and wouldn't be a meaningful way to describe an 80% chance of guilt in a SINGLE case. How can you POSSIBLY be considering cases with DIFFERENT evidence when there is exactly ONE defendant with ONE set of evidence?

[/ QUOTE ]

You may not realize it, but you are now objecting to Sklansky's Model. In that model there are some cases with the Shoe Size fit and some without it. That's how he computes his result.

As far as your other comments, don't expect another reply by me.

PairTheBoard

jason1990

05-26-2007, 08:03 PM

[ QUOTE ]
Do you agree with me that PTB is seizing on your technical objections as a way to justify assertions that far go beyond your points? Sort of like how religious people think erroneously that quantum theory backs up their ideas.

[/ QUOTE ]
Personally, I do not see that, although I admit I have not read all his posts, so I do not know all of his assertions. This business first came to my attention when I saw him taking issue with your argument involving 90% of people disagreeing with a certain theory. For me, personally, if I see a theory where 90% disagree, it sometimes gives me more incentive to take a close look at it. Especially if the theory continues to be discussed despite that. I also make a conscious effort not to be biased in my evaluation of the theory. So I intentionally try to avoid coming to the conclusion you reach.

I do not think a probabilistic argument involving the space of all possible theories is a good way to approach the question. But I recognize that as your own subjective approach; if it helps you and/or others to clarify your thoughts, then great. My own subjective view is that it is the wrong approach. However, I would vehemently object if anyone tried to claim that your approach was an objective and scientific way to answer the question. I think this is the extent of PTB's assertions on this matter, but I may be wrong.

[ QUOTE ]
Do you agree that the vast vast majority of the people who are claiming that independent evidence should not have a great effect on probability of guilt even when the evidence is rare are saying that out of confusion rather than because I am assuming some indifference principle that they would have not trouble assuming themselves?

[/ QUOTE ]
Not really. I agree that the majority could be convinced to accept the indifference principle. I also agree that the majority would be confused by Bayes Theorem and probability theory in general. But I think most people that disagree are not thinking about the evidence in terms of probabilistic models. They are doing a subjective weighing of the evidence in non-mathematical terms. They have reasons for their judgements that they could describe in ordinary language. Some of them might be confused. Some of them might not. I see no reason to think that the vast vast majority are confused.

By the way, I do think that mathematical models can be helpful in these juror-type situations. I think math, in general, is underused and a lot of people's informal analysis would be greatly improved by thinking more mathematically. But I also think there are instances where mathematical analysis can lead us astray. It is sometimes very difficult to see the hidden assumptions in a mathematical model, and a strict adherence to some set of calculations can at times be vastly inferior to an informal analysis in more "human" terms, so to speak. As a case in point, I would refer to my earlier post showing that the size 14 shoe would not only be enough to convict the defendant all by itself, it would be enough to override overwhelming evidence in favor of his innocence. In an extreme case like this, my own (subjective) view is that we would be entirely justified in throwing out the model completely. If I were on such a jury at that point, I would be more interested in listening to people's non-mathematical reasonings than in continuing to discuss these calculations.

[ QUOTE ]
PS What epsilon delta are you talking about?

[/ QUOTE ]
Link (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=8896278&page=0&vc=1) .

jason1990

05-26-2007, 08:48 PM

You appear to have some kind of attitude problem. Congratulations on becoming the second person on my ignore list. Before I start ignoring you, I will give you this one reply for the benefit of other readers.

[ QUOTE ]
Jason, all you did was jump through completely unnecessary hoops, divide all sklansky's numbers by 100 for no necessary reason, and then come up with the same answer.

DUCY?

Since you obviously don't, I'll spell it out for you.

[/ QUOTE ]
You have no idea what you are talking about. I would tell you to go back and think about it some more, but I do not think you will do that.

[ QUOTE ]
Imagine a universe of 1 million people, 800k actual murderers, 200k actual innocents, which is the distribution specified in the problem.

[/ QUOTE ]
Do you even know why we are imagining this universe? It is supposed to be the universe that exists in our minds before we learn about the shoe print.

[ QUOTE ]
Select a person at random. You have a 20% chance of picking an innocent.

Now throw out everybody without the proper shoe size. All the (800k) actual murderers have it.

[/ QUOTE ]
Actually, they do not. These are 800k murderers from the universe in our minds before we learned about the shoe print. Murderers in that universe do not all have size 13 shoes.

[ QUOTE ]
Your argument is to imagine a universe of 1 million people that includes murderers without shoe size 13 (did you even read the problem?)

[/ QUOTE ]
I read the problem and David's solution. Apparently you read only David's solution.

[ QUOTE ]
Your universe, since you can't read the problem, is composed of 990k people =99% people WHO HAVE NOTHING TO DO WITH THE QUESTION SINCE THE MURDERER IN THEIR CASE ISN'T SIZE 13.

[/ QUOTE ]
Thank you for stating the obvious. This is exactly how conditional probability works. Before we get the new information, we have a whole universe of possibilities. Many of them are irrelevant, but we do not yet know that. Once the new information comes in, we are able to determine what is irrelevant. We remove it and compute a new probability.

[ QUOTE ]
There's absolutely no reason whatsoever to consider these people in the context of sklansky's question. It's just jumping through an additional hoop to throw them out (divide both numbers by 100) to get the EXACT SAME answer becuase sklansky's method, for the problem he described, is correct.

[/ QUOTE ]
David gets the right answer for the wrong reasons. The mistake is small and subtle, but anyone who is genuinely interested in learning would benefit from trying to understand it. I would tell you to go back and read the entire thread, but you probably will not do that either. Good luck figuring this stuff out.

TomCowley

05-26-2007, 10:01 PM

You're making the same mistake PTB is. I understand conditional probability perfectly well. When P(guilty|size 13)/P(innocent|size13) = P(guilty|size!=13)/P(innocent|size!=13) = P(guilty)/P(innocent), before any shoe evidence is introduced, which is an inherent assumption in Sklansky's question (you've questioned the validitiy of this assumption, which is fine, since there could be a small correlation, but you then use THIS EXACT ASSUMPTION in your solution), deriving a solution that examines size!=13 is unnecesarily complex. Furthermore, criticizing a solution for skipping that step, WHEN IT IS AN ASSUMPTION IN THE PROBLEM, is idiotic.

Sklansky assumes the equality I have above. You assume it, and then state the only correct solution is one that doesn't use it at step 1 to shortcut some math. W T F?

MegaloMialo

05-26-2007, 10:11 PM

David you are ugly as hell

PLOlover

05-26-2007, 10:13 PM

[ QUOTE ]
If you take a random black, with no evidence at all, out of a population of 1 million (10% black), his "before" guilt is 1/1 million (somebody did it). Knowing a black did it simply makes his guilt 1/100000, which is the exact likelihood of picking that black out of the population of blacks, which is what you should expect hsi guilt to be with no other evidence. There's nothing fishy going on.

[/ QUOTE ]

any prosecutor can whip up some lame evidence to make it 50-50. No prosecutor is gonna get up there and say it is 1000000-1 that this guy is guilty except for this 1 piece of evidence.

PLOlover

05-26-2007, 10:16 PM

[ QUOTE ]
Any "perfectly black or white" evidence where the guilty/innocent must match/not match, such as shoe size, isn't subject to cherry picking. If there were ANY evidence of this type that didn't add up, it's an instant acquittal. That's what pre-trial disclosure is for. If you want to argue about likelihoods of corrupt prosecutors coupled with bad representation, go somewhere else. That isn't what this question is about.

[/ QUOTE ]

you're overlooking a big point, the police/prosecutor decides where the investigation goes, they decide what evidence to collect. It's not like a full investigation is done and all leads are followed equally. Once a suspect is found generally only the leads for that suspect are followed.

PLOlover

05-26-2007, 10:20 PM

[ QUOTE ]
Quote:
If you take a random black, with no evidence at all, out of a population of 1 million (10% black), his "before" guilt is 1/1 million (somebody did it). Knowing a black did it simply makes his guilt 1/100000, which is the exact likelihood of picking that black out of the population of blacks, which is what you should expect hsi guilt to be with no other evidence. There's nothing fishy going on.

any prosecutor can whip up some lame evidence to make it 50-50. No prosecutor is gonna get up there and say it is 1000000-1 that this guy is guilty except for this 1 piece of evidence.

[/ QUOTE ]

Also a better example was the 97% innocent man who has a size 13 the same as evidence so that shoe size swings it to 95% guilty.

TomCowley

05-26-2007, 11:25 PM

[ QUOTE ]
any prosecutor can whip up some lame evidence to make it 50-50. No prosecutor is gonna get up there and say it is 1000000-1 that this guy is guilty except for this 1 piece of evidence.

[/ QUOTE ]

So it's a coinflip for guilty turned into a 10/11 chance of guilt. That's fine. Cases are made from way under 50% on DNA evidence. The rarer the specific matching evidence, the less other evidence you need to get the same likelihood of guilt. It's pretty cool.

[ QUOTE ]
Also a better example was the 97% innocent man who has a size 13 the same as evidence so that shoe size swings it to 95% guilty.

[/ QUOTE ]

It's actually just ~75% guilty (3/(3+0.97)). And DNA can make even bigger swings, and nobody bats an eye. I'm not sure what the psychological block is here.

[ QUOTE ]
you're overlooking a big point, the police/prosecutor decides where the investigation goes, they decide what evidence to collect. It's not like a full investigation is done and all leads are followed equally. Once a suspect is found generally only the leads for that suspect are followed.

[/ QUOTE ]

Jason makes a similar point, and while it's true in a general sense, it's not applicable to the intent of the problem.

It is implicit in the problem that the shoe size is independent of all other evidence. Let's say the "new" evidence was "a right footprint". By itself, this is obviously meaningful. If previous evidence included "7 left and 6 right footprints from the same person/shoe", it's all but meaningless.

Why beat around the bush with biased selection, etc. If you want to introduce correlated evidence, just introduce 13 other copies of the footprint. It's a trivial objection. Skalsnky should have stated that the evidence was uncorrelated, but unless you think he wanted to address my case (another copy of the same footprint), it's safe to assume that it is.

PLOlover

05-26-2007, 11:33 PM

[ QUOTE ]
Quote:
any prosecutor can whip up some lame evidence to make it 50-50. No prosecutor is gonna get up there and say it is 1000000-1 that this guy is guilty except for this 1 piece of evidence.

So it's a coinflip for guilty turned into a 10/11 chance of guilt. That's fine. Cases are made from way under 50% on DNA evidence. The rarer the specific matching evidence, the less other evidence you need to get the same likelihood of guilt. It's pretty cool.

[/ QUOTE ]

DNA match is 1/mill at least, not 1/10.

I guess common sense just tells me that under the model DS proposed and you endorse, being able to convict 10/11 black men with no real evidence just seems wrong and to me points to some underlying flaw in the model.

TomCowley

05-26-2007, 11:38 PM

If "no real evidence" can lead to 50% guilty, that's a far bigger indictment of the legal system than the model. I mean, can you pick any random crime and random person, let the prosecution and a competent defense go at it, and have a jury say that he's just as likely to be guilty as not (not necessarily convicted, but guilty)? If that's true....

PairTheBoard

05-27-2007, 03:10 AM

I'm not sure the Sklansky Model can even handle "Presumed Innocence". It's been said that the Shoe Size evidence might be enough to convict by itself. I'm not sure the Sklansky Model implies this. In fact, I think the Sklansky Model would give No Weight to the Evidence from a 0 probability of Guilt level, much less move the line to 99% guilty.

Suppose we presume just a little guilt. Say we assume the defendant is from the same city of 1 million people as the murder. Then our imaginary group of defendants are the 1 million people in that city and the defendant is judged to have 1 chance in a million of being the murderer. Now the Shoe Size evidence is presented. That cuts the popluation of the imaginary model down to 10,000 which includes the defendant and the murder. That means the line has been moved from 1 chance in a million to 1 chance in 10,000 of Guilt. Not even close to 99%.

But that is not a real presumption of innocence. A real presumption of innocence should start out with 0% Guilt, not 1 in a million. So lets move it closer to 0%. Let's presume the defendant is from the same continent of 1 billion people as the murderer. After the Shoe Size evidence he then has 1 chance in 10 million of Guilt. Practically nothing. In fact, as we move the starting level for guilt closer and closer to 0% where it should be, the Shoe Size evidence Guilt also goes to 0%.

Presumption of Innocence under Sklansky's Model would mean that noone could ever be convicted on circumstantial evidence. Even if you had a chain of such evidence which Sklansky claims would parlay guilt up to 100%, if we start with a presumed innocence level of 0% none of the circumstantial evidence would move the line.

PairTheBoard

PLOlover

05-27-2007, 03:56 PM

[ QUOTE ]
If "no real evidence" can lead to 50% guilty, that's a far bigger indictment of the legal system than the model. I mean, can you pick any random crime and random person, let the prosecution and a competent defense go at it, and have a jury say that he's just as likely to be guilty as not (not necessarily convicted, but guilty)? If that's true....

[/ QUOTE ]

well if that's true then I'm sure every jury would acquit as they definitely have reasonable doubt, as any competent defense attorney would tell them like 50 times.

I would imagine most jurors start out biased in favor of the prosecution whether they realize it or not, so starting out 50-50 is sort of utopian in itself.

Schmitty 87

05-28-2007, 02:14 AM

I think that my example is an accurate application of Sklansky's model, though correct me if I messed up.

Assume a jury is presented with 100 defendants. The jury is told that only one defendant is guilty. So, for defendant X, P(guilty)=.01. Now, the jury is told that the guilty defendant has characteristic Y and that defendant X has characteristic Y. Characteristic Y occurs in .1% of the population. P(guilty) for defendant X will obviously skyrocket. If my calculations are correct, P(guilty) for Defendant X will go up to .91. And that seems fair; after all, 1 of the 100 is both guilty and has characteristic Y, and it is highly likely that that 1 person is the only one who has characteristic Y.

I think that the problem with the model is the disconnect between:
1) the jury being told that 1 of a group of 100 people is guilty
2) the jury coming to the conclusion that a person has a 1% chance of being guilty

In case #1 P(guilty)=.01 is unquestionably reliable, but in case #2 P(guilty)=.01 is unquestionably unreliable. But, when we apply Bayes Rule and all that stuff, both cases yield the exact same results (because we assume the jury's conclusion to be just as accurate as case #1).

PairTheBoard

05-28-2007, 05:42 AM

I can't follow which case is which in your explanation. Let me try to clarify what I think you're getting at.

Case 1.
First, suppose the Jury is told the defendant was chosen randomly from a pool of 100 people where it is known one of the 100 is guilty. At this point, P(Defendant is guilty)=.01

Second. Suppose further, the Jury is told the defendant and the murderer both have characterisic X which is common to only 1% of the general population.

What is the Bayes recalculation of probabilty of guilt?

1/(.99+1) = about 50%

So you're right. It has skyrocketed.

Now consider a completely different example.

Case 2.
First, the Jury knows nothing about the defendant except that he and the murderer have characteristic A which is commonly shared by 1% of the General Public.

Second, the Jury is now told that the defendant and the murderer both have characteristic X which is commonly shared by 1% of the general population.

Now, in Case 2 is the Baysian calculation for probability of guilt the same as Case 1? A lot of people would say so without thinking it through. But it's far different. In Case 2 the jury does not know there is 1% chance of guilt after the informaation about characteristic A. That wouldn't make any sense at all. The jury doesn't know anything about the pool of people the defendant was chosen from.

That's the whole problem. How to quantify presumption of innocence for getting these Baysian caluculations started. 0% doesn't really make sense as a baseline because that would mean the defendant is simply innocent and no evidence should convict him. What is the proper baseline?

If you look at presumption of innocence as being like the defendant was chosen randomly from the 1 million population of the city containing the murderer, then the baseline is 1 chance in a million. That would mean Characteristic A boosts it to about 1 chance in 10,000. And the Second evidence of Characteristic X boosts it to about 1 in 100 - assuming the characterisics are independent. That's 1%. Much smaller than the 50% in Case 1.

This whole idea of getting Jurors to convert their judgement to numbers or even a just a Final Verdict Number is going to lose in translation what is already working fine when the Jurors are instructed on Reasonable Doubt and Presumption of Innocence in Human terms.

PairTheBoard

Schmitty 87

05-28-2007, 02:49 PM

Case #1 is when the jury is told that the defendant has been pulled from a group of 100, one of whom is guilty. For a given defendant, P(guilty) is clearly .01.

Case #2 is when the defendant is not pulled from that group. The jury decides (through other evidence or speculation or whatever) that P(guilty)=.01.

The interpretation of the probability in Case #2 will be the same as the given information in Case #1, that 1 of 100 of a group of defendants will be guilty. But it is clear that the jury's conclusion of P(guilty)=.01 in Case #2 is quite different from Case #1. If I say I am 99% sure about something, what does that really mean? 99%? 99.9%? 99.99%? Those seemingly slight differences are going to factor in huge when we apply Bayes Rule. And how reliable is my conclusion in the first place?

Sklansky says that the way we quantify pieces of evidence is by applying Bayes Rule, and that would work beautifully if we had the information presented in Case #1. However, we're working with Case #2, in which the starting point P(guilty), as you say, CANNOT begin at .01. That would allow the conviction of basically everyone who shared a rare characteristic with the actual murderer.

So yeah P(A given B)=[P(B given A)P(A)]/[P(B)], where P(A)=P(guilty) and P(B)=P(characteristic X). The problem is in finding a reliable initial P(A), as you say.

Schmitty 87

05-28-2007, 02:55 PM

[ QUOTE ]
This whole idea of getting Jurors to convert their judgement to numbers or even a just a Final Verdict Number is going to lose in translation what is already working fine when the Jurors are instructed on Reasonable Doubt and Presumption of Innocence in Human terms.

[/ QUOTE ]

Yeah I suppose, though the idea of facing a jury of my fellow citizens still scares the living [censored] out of me. I wouldn't mind a little something more than feeling guiding their decision, though I imagine that's all that would factor in anyways and they'd just express it in numbers rather than human terms.

PairTheBoard

05-28-2007, 03:31 PM

[ QUOTE ]
[ QUOTE ]
This whole idea of getting Jurors to convert their judgement to numbers or even a just a Final Verdict Number is going to lose in translation what is already working fine when the Jurors are instructed on Reasonable Doubt and Presumption of Innocence in Human terms.

[/ QUOTE ]

Yeah I suppose, though the idea of facing a jury of my fellow citizens still scares the living [censored] out of me. I wouldn't mind a little something more than feeling guiding their decision, though I imagine that's all that would factor in anyways and they'd just express it in numbers rather than human terms.

[/ QUOTE ]

It's the translation to Numbers I'm worried about. Especially the determination of Reasonable Doubt. Look at the Human terms the court puts it in.

Reasonable Doubt -
"Any doubt which would make a reasonable person hesitate in the most important of his or her affairs."

Now look at the the responses people gave when Sklansky asked people to translate that into a probability of Guilt. He did this in another thread. The Numbers are all over the place. The best Number given by a couple of posters, ZERO, was mostly scoffed at. Yet numbers like 95% were considered reasonable. Why not just leave it in the human terms above that people understand.

If I am getting ready to cross a bridge - an important matter for me if it's unsafe - and I know that the bridge is going to fail and I'm going to die 1 time in 20 that I cross it, do you think I just MIGHT Hestitate to cross it?

That's the kind of thing the court wants people to think about when they have the Life of the Defendant on the line. That's the kind of thing I want the Jurors to think about if they have My Life in their hands. Not some high falootin theoretical Number.

PairTheBoard