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komponisto comments on The Amanda Knox Test: How an Hour on the Internet Beats a Year in the Courtroom - Less Wrong

42 Post author: komponisto 13 December 2009 04:16AM

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Comment author: komponisto 15 January 2010 08:59:34PM *  5 points [-]

A flurry of more recent comments, (concerning in particular the nature of evidence), plus some private message correspondence, provides me with an excuse to make what are probably somewhat-overdue comments succinctly summarizing the main points of this post, addressing the most important issues and objections raised by others, and tying up some loose Bayesian ends.

Objections raised to my arguments seem to fall mostly into the following categories:

(1) Bayesian pedantry: pointing out that weak evidence is distinct from zero evidence, that rational evidence is distinct from legally admissible evidence, arguing that I chose "the wrong prior" (as opposed to just coming out and accusing me of failing to update on piece of evidence X), etc.

(2) Claims that I am too dismissive of psychological evidence.

(3) Object-level arguments from a few people who (still) think there is a significant chance Knox and Sollecito are guilty.

Here are some general responses, divided up into three separate (threaded) comments for ease of reading (and possibly replying):

(1) Bayesian pedantry.

In the first version of the post, when I wrote "there is for all intents and purposes zero Bayesian evidence that Amanda and Raffaele are guilty", I thought that "for all intents and purposes" was enough of a disclaimer; but really, of course, what I should have written was: "there is for all intents and purposes zero net Bayesian evidence...". Obviously there is some (weak) evidence of K and S's guilt; but (I claim) it is cancelled out by the evidence of their innocence to such an extent that the posterior probability is close to (not necessarily literally equal to) the prior probability.

And yes, by "prior probability", I am entitled to mean the probability that Amanda Knox would commit murder, rather than the probability that Amanda Knox would commit murder given that a murder occurred in her house. I have not neglected to take into account the latter information, and indeed, the fact that her housemate was murdered obviously raises the probability that Knox is guilty of something to a level significantly above the background prior. However, this probability is lowered again by the information that Guede committed the murder -- and this is true despite the fact that evidence of Guede's guilt is not itself evidence of Knox's innocence. As Eliezer put it,

the fact that we know Guede did it means that there's no unexplained murder around for Knox to be convicted of.

The Bayesian structure of this inference is simple enough. Let H be the hypothesis that Knox killed Kercher, and let E be the datum that Kercher was killed. Obviously, H and E are not independent; in fact, H strictly implies E, which means that P(H&E) = P(H). Thus P(H|E) = P(H)/P(E), which is larger than P(H). Now, however, let E be the datum that Kercher was killed by Guede. Then H and E are much closer to being independent, i.e. P(H&E) is only slightly larger than P(H)P(E), which means that P(H|E) is close to P(H)P(E)/P(E) = P(H). As I argued in the post, this is Occam's Razor at work: P(H&E) = P(H)P(E) << P(E).

To those who gasped at my dismissal of the DNA evidence against Knox and Sollecito when it is my solemn Bayesian duty to consider all available information, I would put it that Bayesian rationality is supposed to be more accurate than legal reasoning, not less. The kind of legal rules of evidence that would come into play here exist because of known biases people have toward convicting defendants (others, not relevant here, exist as a reflection of society's willingness to let a few culprits go free in order to maintain a certain incentive structure on police and prosecutors). Thus, if when you put your Bayesian goggles on, you suddenly find that defendants are looking more guilty than they used to, you're probably doing something wrong. In any case, dismissing evidence is not an illegal Bayesian move; it simply constitutes an assertion that the likelihood ratio for the information in question is close to 1. In other words, the evidence is very weak. Which probably has something to do with why it might not be admissible in court.

Comment author: komponisto 15 January 2010 09:01:58PM *  4 points [-]

(2) Claims that I am too dismissive of psychological evidence.

Some folks appear to have misunderstood me as arguing that psychological evidence can be dismissed out of hand by its very nature, or that Amanda Knox shouldn't have been questioned by police despite being Kercher's roommate. Obviously, neither is the case. First of all, psychological evidence, strictly speaking, is physical evidence. In fact, all evidence is physical. The question, always, is about how strong a given piece of evidence is. In the post, I mentioned two rules for this: (i) the closer the spatiotemporal connection, the stronger the evidence -- something which follows from the laws of physics; and (ii) high-prior data is at most weak evidence of a low-prior hypothesis (or: thou shalt not postulate a strange thing to explain a less strange thing) -- an immediate consequence of Bayes' Theorem.

If a person's behavior were sufficiently unusual, it might constitute significant evidence they were involved in a homicide. And if a murder occurs in Knox's house, Knox is clearly one of the first people the police should interview. But the psychological evidence against Knox is in fact very weak: the causal linkage between Knox's cartwheels, phone calls, or speculation about Lumumba's involvement on the one hand, and a hypothetical murder committed by her on the other is very tenuous compared to the causal links between these things and more local factors which are likely to be largely independent of Kercher's murder. And yet, many people mistakenly believe the evidence is strong. Why? Because, for perfectly understandable evolutionary reasons, people are interested in psychology; it's salient in their minds, and they mistake this salience for evidentiary strength. They find it easy to imagine a guilty Amanda Knox doing cartwheels, and caught up in the narrative, forget to properly consider the ways in which an innocent Amanda Knox could also end up doing cartwheels. (In fact cartwheels were never something particularly unusual for Knox.) And, in continuing to consider such weak evidence against Knox after the strongest evidence in the case -- the evidence from Kercher's room -- has pointed decisively to the crime having been committed by Rudy Guede, they compound the error by privileging the hypothesis, or, in other words, overestimating the prior probability, of Knox's guilt.

Comment author: komponisto 15 January 2010 09:05:37PM *  4 points [-]

(3) Object-level arguments from a few people who (still) think there is a significant chance Knox and Sollecito are guilty.

There don't appear to be many LW regulars who assign high probabilities to Knox and Sollecito's guilt. There are a few who remain apprehensive about doubting the rationality of the jury or of people like Judge Micheli (the judge in the Guede case) to the extent of lowering the probability of guilt to single digits of percentage; but mainly, they agree that acquittal is called for. My remarks here will thus be directed mostly at people chancing upon this post from outside.

It is incorrect to assume, as some people have, that because I didn't mention various facts about the case in this post, that I am therefore unacquainted with them. On the contrary. However, I do assert that the essence of the case -- the decisive information -- is just what I presented here. Among my main points was that people have been distracted by various other details to the point of getting the wrong answer. The central case in point would be the eight members of the Perugian jury, who despite hearing all kinds of information over the course of nearly a year, were unable to sort out the clutter from the essential facts, which are these:

  • The prior probability of Knox and Sollecito having committed murder is extremely low; of having committed this particular kind of murder, even lower.

  • Since Guede is unquestionably responsible, the mere fact of Kercher's death does not need to be explained.

  • There is no evidence that Knox was in Kercher's room on the night of the crime, and almost no evidence that Sollecito was.

  • There is almost no evidence of any association between Knox and Guede, still less of any between Sollecito and Guede.

  • There is no evidence of antipathy toward Kercher on the part of Knox or Sollecito, or of any other plausible motive for the crime.

  • Every piece of apparently strong prosecutorial evidence has either been totally discredited or is subject to serious doubt (examples: footprints, "mixed" blood, knife, bra clasp, cleanup,...).

  • The remaining prosecution case consists of odd physical facts that are nowhere near as strange as Knox or Sollecito being guilty (broken window circumstances, apparent rearrangement of scene), dubious psychological inferences (inconsistencies => cover-up, cartwheels => not sad, not sad => guilty), and outright non-sequiturs (Knox named Lumumba & Lumumba innocent => Knox hiding something).

  • Knox and Sollecito (along with Lumumba) were suspects before Guede was; the basic theory of the crime was not revised when Guede was discovered.

The above facts are indisputable; there can be little question about which way the winds of evidence blow in this case. Yes, there are lots more detailed facts you can learn about the case (and I know a bunch of them), but as regards the question of who is guilty and who is not, the above is what you need to know, and the rest is noise.

Comment author: rolf_nelson 01 February 2010 08:41:25PM *  -1 points [-]

Point-by-point:

Given that AK's roommate is dead, a break-in was staged, and the coroner's report showed multiple attackers, the prior on AK being a murderer of Meredith is rather high. On the other hand, if we throw away all known evidence, the prior of AK (or Guede, for that matter) being a murderer of Meredith is less than one in a trillion. I claim the former approach, where you use evidence rather than ignore it when it's inconvenient, is preferable. [Edit: OK, that was too snarky. Let me instead say that you should start with a tighter prior rather than a looser prior where possible; it makes the math more tractable.]

That said, I think our main disagreement is on whether the prosecutorial evidence holds up.

  1. Why do you believe DNA evidence flies around so easily? Quick tests: Do you find your beliefs about DNA match up with how DNA is used to draw conclusions in any other court cases that you're familiar with? Why was RS and the other roommate's DNA not found in more areas? Google any video of a DNA testing lab. Are they wearing hazmat suits and, if not, why aren't the testers contaminating their own samples left and right?

  2. I disagree the five pieces of evidence you listed (footprints, DNA-mixed blood, knife, bra clasp, cleanup) are discredited. I am interested in hearing why you personally believe each of them is not strong evidence. [Edit: if you limit yourself to one item, my order of preference is DNA-mixed blood, then knife, then bra clasp.)]

  3. Outside of those five pieces of evidence, I also think you're being too dismissive of the other pieces of evidence. For example, why is Knox naming Lumumba a non-sequitur?

  4. If the basic theory was that multiple people, including at least one roommate, killed MK, then there was no reason to abandon that basic theory upon discovering Guede.

Am I correct in that one of our disagreements is this:

Observation: AK claimed she saw Lumumba kill Meredith. Lumumba was therefore detained by the police based on her affidavit, but Lumumba turned out to be innocent.

Your conclusion: The presence of this false accusation decreases the probability of AK's guilt (?!) (Because it somehow leads credence to a railroading theory? You may be buying into a false FoA meme that the prosecution merely replaced Lumumba with Guede to save face.)

My conclusion: The presence of this false accusation increases the probability of AK's guilt.

Comment author: Jack 01 February 2010 11:48:53PM 9 points [-]

My ADHD brain lost interest in this after the huge discussion here covered pretty much everything having to do with the trial. And komponisto knows this stuff way better than I do so I'll let him respond. But nearly all of the regulars here looked at the exact same evidence you are bringing up now and nearly all of us eventually concluded that it was more likely than not that AK was innocent. Moreover, I specifically recall discussions of every single point you bring up and I believe every single one was resolved. So you might want to look back at the comments to the original post (the test of your rationality one) before you make komponisto go through all this again.

Since I was heavily involved in the discussion of the Lumumba 'accusation' I'll just add this: If AK and RS cleaned up their own involvement with the crime, but intentionally left the evidence implicating Guede wtf would AK not just accuse him to begin with? It seems likely bordering on obvious that AK was interviewed under duress, that the police had seen her text to Lumumba as implicating evidence and fed her his guilt. As a result, AK either made up or dreamed this bizarre pseudo-memory in which Lumumba was present. Then when Lumumba was cleared and an actual suspect was found they kept AK in their hypothesis so as to not totally embarrass themselves. Indeed, this explanation is one of the few parts of the story where I feel like I can account for every single fact and where no other particularly different explanation makes sense. Even if Knox did kill Kercher I would still assign this kind of story a high probability.