It seems that the Italian judicial system is indeed applying Bayesian inference... using a Monte Carlo approximation :)
This makes for a good case study in base rate neglect. Girls like Amanda Knox almost never commit the type of murder she was accused of, so unless you have extremely compelling evidence you should think she is very likely innocent. Yet would we want our justice system to grant a higher presumption of innocence to pretty, smart, young women, then, say, lower class men?
No. This was already discussed here.
If she had been randomly sampled from the general population then the prior probability would have been exceedingly small, but she wasn't randomly sampled, she was investigated because she lived with the victim. When someone is murdered, there is a high probability that the perpetrator is somebody in a close or frequent relationship with them.
Realistically, the prior probability of Knox being a perpetrator would be around 0.01, the same for Sollecito, with high positive correlation between them.
That's still base-rate neglect as you are picking and choosing what you want to look at and not conditioning on one of the more relevant variables.
What fraction of the pretty girls who lived with the victims turned out to be murderers? By looking at the genderless conditional probability ('somebody'), you're implying that women like Knox might have male-like murder levels, which is obviously wrong. And to the extent that pretty girls do not have differing patterns of murdering roommates from other women, you're making the exact same mistake, even (it doesn't matter whether you update on pretty girl then roommate or roommate then pretty girl).
Update on both living with the victim and being female and the small probability is bigger but... still small, since the still relatively low probability of a roommate murdering is penalized substantially by being female (female murder rates are like 1/10th male and that's the raw rates, not adjusted for age or SES or race etc). As the top comment says, "Once we take into account that AK and MK aren't in a relationship, AK is female, and there is very strong evidence that someone else committed the murder then I'd agree that the probability drops".
Isn't this also confounded by the fact that judges and juries like to go easy on women, so that women who do commit murder are less likely to be convicted? It may be that measures of what fraction of women are convicted of murder are not the same as what fraction of women are actually murderers.
Most statistics are based on police reports, not convictions. For crimes other than murder, there is a good agreement between police reports and surveys of victims.
Conviction data has a bias introduced by the the court, but it has a much worse bias from restricting to cases where a suspect is identified and apprehended.
Prosecutors may also be less likely to accuse women. I wonder what is the female rate of being accused of murder - if it is 1/10 just as the murder rate is, then this 1/10 can cancel out in the courtroom.
The prosecutor is already using what ever priors they wish, including racist and sexist priors, when they select the suspects to bring to the court; if the court is to do the same, they'll be double-counting.
Ultimately it's all in the wash once you start accounting for things like her trying to frame Lumumba.
Keep in mind also that there's evidence available to prosecution but unavailable to you. Knox claiming that she got slapped during interrogation, and other claims that those present at the interrogation know for certain to be true or not.
I can see it going either way: if I were the police present at the interrogation and then I see her completely lying about how interrogation went, then the reference class is not cute girls it's psychopaths and not very smart ones either. On the other hand maybe she didn't lie about the interrogation. I can't know but those present at the interrogation would know.
edit: also the thing is that a lot of the physical evidence was not reported on by the US media.
Basically there is a lot of physical evidence that if valid would massively overpower any "cute girl" priors. So the question is not about those priors but about the possible alternative explanations for said evidence and said evidence's validity.
Keep in mind also that there's evidence available to prosecution but unavailable to you. Knox claiming that she got slapped during interrogation, and other claims that those present at the interrogation know for certain to be true or not.
Even if the lied that she was slapped, that doesn't suggest that she's guilty. It rather shows that she was under a lot of pressure which is expected. It doesn't make someone a psychopath to break under strong pressure and being accused of murdering your roommate is strong pressure.
Well it's a fairly specific type of breaking down, to be accusing other people. There's other ways of breaking down, you know. And if her account of interrogation is false, and the police's account is true, that goes well beyond the lie about slapping. She said she was at the scene of the crime covering her ears as black owner of the bar she works at was murdering the victim, and if you know you didn't coerce the witness into making such a statement, that's very different from coercing a witness into such a statement.
While perhaps insufficient evidence in the court of law, the prosecutor is not the court of law, the prosecutor merely needs a strong suspicion for it to be their job to try to convict.
Ultimately we have Knox's words against the police's, and both sides have a coherent story that makes either side right.
Well it's a fairly specific type of breaking down, to be accusing other people. There's other ways of breaking down, you know.
Yes, signing a confession would be another typical one. In that case she would have it even worse.
Her being psychopathic would have likely lead to other facts that a well funded persecution could uncover.
I think it'd be quite strange to claim that confessions don't ever correlate with guilt.
By the way what she did was she claimed she was at the scene of the crime covering her ears as Lumumba murdered Kercher (and no she didn't call the 112 about it or anything). If she as she says was coerced into making such a statement, yeah, that's not evidence of guilt. But if it is as police says it is, do you still think it's not evidence of guilt?
Picture an alternative universe. Bob, an exchange student from Australia, is being questioned as a witness. There's a minor discrepancy: Jake, his friend, withdrew his alibi for the night. Those things happen, you don't really think too much of it, but you have to question Bob. You're somewhat suspicious but not highly so. Without much of a prompt, Bob tells a story of how he was covering his ears as Peter, his boss, was murdering the victim.
Now what do you do with Bob, exactly? Let him go once you clear Peter? Keep him because he's not a cute girl?
Now, we aren't sure that this is how it went. Police claims that this is how it went and Knox claims that she got pretty much beaten into that statement, and it's one word against the other.
Her being psychopathic would have likely lead to other facts that a well funded persecution could uncover.
She's a foreigner, there's no budget for transatlantic flights to figure out if she had been cruel to animals as a child or the like, there's no jurisdiction, and you can't use that sort of stuff in a court anyway.
By the way what she did was she claimed she was at the scene of the crime covering her ears as Lumumba murdered Kercher (and no she didn't call the 112 about it or anything). If she as she says was coerced into making such a statement, yeah, that's not evidence of guilt. But if it is as police says it is, do you still think it's not evidence of guilt?
The police records indicate that they had already started considering Lumumba as a suspect prior to interrogating Knox. Knox was detained for a long period of time by the police, during which time she alleges she was treated abusively, before she pointed her finger at the person the police already suspected, but who later proved to have an airtight alibi.
She's a foreigner, there's no budget for transatlantic flights to figure out if she had been cruel to animals as a child or the like, there's no jurisdiction, and you can't use that sort of stuff in a court anyway.
The prosecution presented plenty of character evidence, the worst they had to present was simply very innocuous, even when they tried to exaggerate it for effect. In the prosecution's hands, an anime series which a member of Less Wrong attested to having watched in an after school club in middle school became a work of "violent animated pornography."
Update on both living with the victim and being female and the small probability is bigger but... still small, since the still relatively low probability of a roommate murdering is penalized substantially by being female
Low compared to what? For someone murdered at home in the dead of night, the dominant probabilities are that either the murderer was invited in or lived there. Roommates merit investigation. If the evidence clears spouses/lovers and close family, then the probability of it being a roommate goes up considerably. Being female is not going to lower the probability enough to eschew a thorough investigation.
What saves Amanda Knox in this case isn't being female, but rather evidence that someone else committed the crime, as well as the lack of physical evidence of her involvement or any paper trail pointing to a conspiracy.
For someone murdered at home in the dead of night, the dominant probabilities are that either the murderer was invited in or lived there. Roommates merit investigation. If the evidence clears spouses/lovers and close family, then the probability of it being a roommate goes up considerably. Being female is not going to lower the probability enough to eschew a thorough investigation.
You're not disagreeing, but you're failing to consider the numbers here. If, say, a quarter of people are murdered by their roommates, and males are 10x more likely to be killers than females, what's the odds of a female roommate doing it?
Being female is not going to lower the probability enough to eschew a thorough investigation.
A probability like 2.5% is worth following up on if police have no better leads to focus on, but they visibly focused on it way more, and in fact people focused on it way more; consider how many expressed probabilities were higher than that in the LW survey. And consider the implicit probabilities in the faction of the public and the Kirchers baying for Knox's blood.
All consistent with base-rate neglect (of being female).
If, say, a quarter of people are murdered by their roommates, and males are 10x more likely to be killers than females, what's the odds of a female roommate doing it?
Depends on whether murder by roommate and murder by female are independent. An average taken over the all homicides includes gang violence, robberies, bar fights, etc. Some kinds of murders are overwhelmingly perpetrated by males, while others are more balanced (for example, males are only 50% more likely to kill their children than females). Once we narrow down the circumstances of the murder, all kinds of dependencies and conditionals start popping up and the base rate becomes less relevant.
A probability like 2.5% is worth following up on if police have no better leads to focus on, but they visibly focused on it way more, and in fact people focused on it way more; consider how many expressed probabilities were higher than that in the LW survey.
Agreed. Police try to shoehorn in their theory and the press isn't going to let the truth get in the way of a good story.
Okay, let's go with your number... let's suppose hypothetically that you aren't beating or otherwise unduly coercing cute girls into saying what you want, and you started with the probability of 2.5%. Then your suspect tells you they were at the house covering their ears not to hear the screams as their big black boss murdered the victim. Now what happens to 2.5%? After you clear the big black boss, what happens?
I don't think you can claim base rate neglect without also claiming police brutality, coercion, and leading the witness (which would be a much bigger problem)
extent that pretty girls do not have differing patterns of murdering roommates from other women, you're making the exact same mistake
That's an interesting claim. Do you have a source for "pretty girls are less likely to murder than girls who aren't pretty?"
Working Paper Ugly Criminals "Using data from three waves of Add Health we find that being very attractive reduces a young adult's (ages 18-26) propensity for criminal activity and being unattractive increases it for a number of crimes, ranging from burglary to selling drugs. A variety of tests demonstrate that this result is not because beauty is acting as a proxy for socio-economic status. Being very attractive is also positively associated adult vocabulary test scores, which suggests the possibility that beauty may have an impact on human capital formation. We demonstrate that, especially for females, holding constant current beauty, high school beauty (pre-labor market beauty) has a separate impact on crime, and that high school beauty is correlated with variables that gauge various aspects of high school experience, such as GPA, suspension or having being expelled from school, and problems with teachers."
More generally: Good human traits are almost always positively correlated with most other good human traits.
But does beauty influence our judgement in accordance with the correlation, or disproportionally so? It may be for example that ugly people are 10% more likely to commit crimes, 200% more likely to be villains in the movies, and 100% more likely to get flagged as suspects by the prosecutor, or get other massive penalty before you even think consciously about it.
Same here. The reason I think so low of the self proclaimed Bayesianism is the sort of thinking where someone sees someone ugly accused and they're like, ha, I am going to be more rational than everyone else today, by ticking my estimate of the guilt up because they're ugly. Completely ignorant that it even makes a difference to the way you should apply Bayes rule that the police and the witnesses and the like had already picked the suspect with this sort of prejudice.
It seems to me that knowing only a little (and/or being bad at applied math) is kind of a pre-requisite for the level of enthusiasm involved in the use of it as a movement name. It's exciting to see all those bits of evidence and see yourself one-upping all those classy educated people that are dead set against use of those bits of evidence, or who even seen to use them in the completely wrong way. It's even more fun to do that with friends.
You know about little math, and it makes a huge difference to everything, that's exciting.
Or you spent years studying and/or working and all that math almost never matters - almost any evidence that's not overwhelmingly strong is extremely confounded with what's already been considered and/or with the chain of events bringing something to your attention.
Both a true correlation and a perception bias may be present, but it would be difficult to distinguish them without using standardized tests.
Correlations between attractiveness and academic performance or criminal record could be confounded by the perception bias, we would need something like IQ or SAT to have a fair estimate.
Also the correlation itself may be caused by perception biases directly, e.g teachers unaware of the halo effect rank the intelligence and agreeableness of the beautiful students greater than they should and such are more unlikely to expel the students or report behavioral problems.
By looking at the genderless conditional probability ('somebody'), you're implying that women like Knox might have male-like murder levels, which is obviously wrong.
No, male-like murder levels would be higher than genderless murder levels. And, if I understand correctly, most of the excess male murder rate involves gang-related violence, which in this case was pretty clearly not involved.
Anyway, I agree that if you are doing pure Bayesian inference you have to condition on all kinds of available evidence, including gender, race, social class, nationality, etc. But we can't expect courts to consider this kind of evidence, for good reasons (avoid creating self-fulfilling prophecies and avoid incentivizing crime within certain demographics).
Right. According to this huffpo post less than 10% of homicides are gang-related, which makes it impossible that gang violence could explain the 10:1 (male:female) homicide offending ratio.
I was thinking that most murders are gang-related and most gang members are male, but I see that this is disputed. Unfortunately, all the sources I can find seem to take a partisan position in the gun control debate, hence I don't know.
Bad prior. Gang violence is a major murder statistic, but it's pretty far from being "most". Quick googling says: "1 in 6 murders". The most common motive, at 50% is "Argument". So.. men are more likely to escalate those to homocide?
Bad prior. Gang violence is a major murder statistic, but it's pretty far from being "most". Quick googling says: "1 in 6 murders".
Makes sense.
No, male-like murder levels would be higher than genderless murder levels.
...and what do you think that implies about whether female murder levels are lower as I claimed?
And, if I understand correctly, most of the excess male murder rate involves gang-related violence, which in this case was pretty clearly not involved.
Yeah, no. Think about that a little bit. (Also, please note the irony of responding to criticism about not conditioning by claiming it would be neutralized by further conditioning.)
(Also, please note the irony of responding to criticism about not conditioning by claiming it would be neutralized by further conditioning.)
If the updates on different kinds of evidence would likely cancel each other, it is an argument for avoiding conditioning too hard or privileging one kind of evidence while doing informal reasoning.
I disagree strongly. The fact that she lives with the victim doesn't shield off the effect of her demographic profile on the likelihood she committed the crime.
Let's analyze the problem using Bayes Net terminology. Let A={suspect=Knox's demographic profile}, B={suspect lives with victim} and C={suspect guilty}. Then your claim is that the net is structured as A->B->C, or that the demographic evidence is conditionally independent of guilt given co-habitation. My claim is that the net is structured as A->C<-B; both A and B affect the likelihood of guilt, and in particular A substantially reduces the likelihood of guilt as James_Miller points out (Note that I'm not saying B is irrelevant, obviously this is wrong).
I am very confident in this claim and would wager long odds in favor of it.
Let's analyze the problem using Bayes Net terminology. Let A={suspect=Knox's demographic profile}, B={suspect lives with victim} and C={suspect guilty}. Then your claim is that the net is structured as A->B->C, or that the demographic evidence is conditionally independent of guilt given co-habitation.
No.
My claim is that the net is structured as A->C<-B; both A and B affect the likelihood of guilt, and in particular A substantially reduces the likelihood of guilt as James_Miller points out (Note that I'm not saying B is irrelevant, obviously this is wrong).
I agree.
.. The thing that puzzles me here is why Knox was ever prosecuted at all. The prosecution had Guede. Who left his fingerprints all over the scene, fled the country, had a history of burglary and knives and changed his story repeatedly. That's a pretty simple and very solid case. Why the heck the prosecution insisted on trying to pin the crime on two more people who could have no plausible reason at all for conspiring with him is just inexplicable to me. I mean, traces of dna from people who lived in the apartment? Wtf? All that proves is that they indeed, lived there.
The thing that puzzles me here is why Knox was ever prosecuted at all. The prosecution had Guede.
The answer is simple and banal: they didn't get Guede until after they had already decided Knox and Sollecito were guilty. Not prosecuting Knox and Sollecito would have required them not only revise to previous beliefs in which they had become psychologically invested, but also to retract previous public pronouncements -- in short, to admit they had been wrong.
From the inside of their minds, no doubt, Knox and Sollecito just felt so suspicious, in the early days of the case before the physical evidence came in and they were relying on behavior to form hypotheses . It's also likely that they were irrationally angry at Knox because of the false implication of Patrick Lumumba that they coerced out of her, and that this anger and frustration at the failure of their own hypothesis morphed into a sense that Knox was an evil vixen.
I think you skip some details. Sollecito withdrew his alibi for Knox. Then Knox implicated Lumumba. And they really go after the guy. Interestingly they fail to railroad Lumumba in the way in which you think they railroaded Knox. Which to me is really interesting because it doesn't fit the 'evil police' story.
Knox of course claims it was extremely coercive, took hours, and some physical abuse from the police. Police denies abuse. We can't really tell either way, but prosecution ought to know how coercive they were. So that's another opportunity to really piss prosecution off.
edit: another thing, wounds and bruises on the body were interpreted as Kercher having been held by one person and stabbed by another. This is the reason why prosecution got so completely sure that more than one person was involved. Yeah, it's rather subjective and unreliable but people can be very sure in that sort of stuff.
There's all sorts of complicated details that are completely missing from the US coverage of the trials, which make the prosecution's position much more understandable. Perhaps the prosecution did not have sufficient evidence, but neither did the prosecution come up with some batshit insane theory out of the blue for no reason when they had everything explained with Guede.
edit: also, Guede was not some random robber, he knew people downstairs and met Knox before at least briefly. If he was random robber who never set his foot on the premises, then Bayesian wise it would have been a no-brainer: it's just unlikely that two independent groups of people who had no chance to pick eachother would be on board with murdering.
There's all sorts of complicated details that are completely missing from the US coverage of the trials, which make the prosecution's position much more understandable. Perhaps the prosecution did not have sufficient evidence, but neither did the prosecution come up with some batshit insane theory out of the blue for no reason when they had everything explained with Guede.
Komponisto is Italian and translated documents from the prosecution for the benefit of the community.
Komponisto is Italian and translated documents from the prosecution for the benefit of the community.
I'm not Italian, just a polyglot.
Selection bias in us hearing about it since a foreign government unjustly prosecuting an extremely attractive American girl is going to generate more publicity than if the girl were seen to be probably guilty.
Nah, pretty girls arrested far from home get covered regardless of guilt or innocence. I've heard about the range of Australians arrested for smuggling drugs into Singapore only in the context of heavy coverage of pretty girls.
Did the prosecution ever even settle on a motive for Knox/Sollecito? I know they had concocted some sort of Satanic sex ritual theory at one point. Was there ever a more plausible follow-up theory for motive?
I'm just surprised to see that the Kercher family is sad that the accused were acquitted.
Why do the Kercher family think that Amanda Knox and Raffaele Sollecito are guilty?
Update: Here's a clue to the family's thinking:
Whether that faith would remain solid after the court of cassation’s ruling, however, was unclear. Although the family have always been careful not to personalise the legal battle, they may well find the definitive clearing of both Knox and Sollecito hard to fathom. An earlier verdict by the court of cassation, which found Rudy Guede, an Ivorian, definitively guilty of Kercher’s murder, specified that the murder could not have been carried out by him alone, and that he must have had accomplices.
How, therefore, the only other people who have ever been seriously considered suspects in the case are now to walk free – for good – as a result of the same court is likely to be a bitter pill for the family to swallow.
An interesting piece of easily quantifiable Bayesian evidence could be phones being switched off overnight (dropping off the network) - how often did Knox do that? If she only did that once in many hundred days, on that night in particular, then that could be a very huge amount of evidence. Or she may have done that few times a week, in which case it's irrelevant.
If she only did that once in many hundred days, on that night in particular, then that could be a very huge amount of evidence.
....No. Not even slightly. This line of questioning MIGHT be relevant if you didn't already have the killer identified, with overwhelming physical evidence pointing towards them. You don't need to explain why Knox turned her phone off, because you already have the killer and every single piece of physical evidence at the crime scene accounted for.
Well, for what it's worth their wounds and bruises guy didn't think it was a single killer. And when someone's murdered at their own place in the dead middle of the night, often the cohabitants are involved.
The prosecutor claimed that from the number of stab wounds, it was unlikely that a single person could have inflicted them all. However, the number of stab wounds was by no means an outlier among murders known to have a single perpetrator (I do not have detailed statistics on this subject, but merely from my limited experience with case studies on the subject I have encountered quite a few cases which involved many more stab wounds from a single perpetrator.) Considering the pervasive incompetence their forensics teams demonstrated over the course of the case, I would assign very little weight to this.
The prosecution also presented pieces of "evidence" such as Knox placing extremely short phone calls to Kercher, too short to transmit any message. This and many other points raised by the prosecution fit the pattern of behavior that seems unusual, and so is presented as evidence for suspicion of murder, despite the fact that the behavior doesn't make more sense if we suppose she was involved in the murder.
If Knox had a murder likelihood of 1/1000 after conditioning on the evidence that Kercher had been murdered, but before accounting for other evidence, and she's then observed to have engaged in unusual actions with a 1/1000 probability, it makes no difference towards her likelihood as a culprit if they're not actions which are more likely in the event that she's actually guilty. We can come up with post-hoc explanations for why the unusual things might be related to involvement in the murder, and this kind of reasoning appears to have constituted a large part of the prosecution's case, but if we don't have any prior reason to suppose that guilt of murder is associated with such behaviors, then these explanations will tend only to be rationalizations of preexisting suspicion.
Here are the New York Times, CNN, and NBC. Here is Wikipedia for background.
The case has made several appearances on LessWrong; examples include: