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 the 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 ket 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.
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 the 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 Bayesian structure of this inference is simple enough. Let H be the hypothesis that Knox killed Kercher, and ket 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.