The point is that the calculation you gave is missing too many steps to be useful for someone just coming to the discussion.
To take another example, does your prior take into account the gender of the suspect? (Females commit far fewer murders than males.) Or is that also screened off by some other evidence?
I suspect that a lot of the details you’re wondering about will quickly emerge in the discussion with Rolf, since (1) our opinions are widely separated, which seems to imply very different-looking calculations, with substantial inferential gaps to be bridged, and (2) that discussion is just getting started.
That said, I’m not sure which missing steps you consider the most important. A very short case summary from my point of view would be something like “student killed by burglar; housemate and boyfriend blamed before burglar discovered; after catching burglar, police filter evidence to fit three-person theory instead of dropping initial idea.” Is that helpful at all?
To take another example, does your prior take into account the gender of the suspect? (Females commit far fewer murders than males.) Or is that also screened off by some other evidence?
A reference class that gives an upper bound for my prior would be “intelligent 20-year-old female college student with no criminal history commits murder”.
At 0.013 per 1,000 people, Italy has the 47th highest murder rate in the world.
Which gives no more than 0.000013 probability that Knox is a murderer if all we know is that she lives in Italy. I guess “intelligent 20-year-old female college student with no criminal history” is less likely to commit murder than average, so I’m still confused how you got “between 0.0001 and 0.001″.
Well, the answer I suppose is that I wasn’t taking the country into account.
However, if you agree that “between 0.0001 and 0.001” is an upper bound, that surely suffices! The important kind of confusion would be where you think my prior is too low, rather than too high.
The important kind of confusion would be where you think my prior is too low, rather than too high.
I was trying to understand what evidence has been taken into account in your prior (i.e., is there some other information that might be considered Bayesian evidence against Knox, but which is already in your prior), so that I can understand what other evidence you consider “negligible”. I think at this point that confusion has been resolved.
I still wonder why the two sides don’t each post a more detailed Bayesian calculation. Let’s say A=”Knox killed Kercher”, B=”Kercher has been killed and Knoxed lived in Italy and is an intelligent 20-year-old female college student with no criminal history”, C=”evidence against Guede”, D=”Knox and Kercher were roommates”, E=”evidence of a staged burglary”, F=”bra and clasp”, G=”all other information about the case”. What are
P(A|B)
P(A|B&C)
P(A|B&C&D)
P(A|B&C&D&E)
P(A|B&C&D&E&F)
P(A|B&C&D&E&F&G)
(Or some other set of evidence and order of evaluation that might be more appropriate.) Wouldn’t that help to quickly pinpoint where your disagreements are?
Let’s say A=”Knox killed Kercher”, B=”Kercher has been killed and Knoxed lived in Italy and is an intelligent 20-year-old female college student with no criminal history”, C=”evidence against Guede”, D=”Knox and Kercher were roommates”, E=”evidence of a staged burglary”, F=”bra and clasp”, G=”all other information about the case”.
I’ll redefine slightly:
A := “Knox killed Kercher, given background info about both, but not the fact of their acquaintance”. P(A) = tiny.
B := “Kercher killed”. P(A|B) = approximately P(A). (We are not yet given that they were roommates.)
C := “evidence against Guede”. P(A|B&C) = approximately P(A). (No significant connection between Guede and Knox.)
D := “Knox and Kercher were roommates”. P(A|B&C&D) = slightly higher than P(A), but still well below the threshold of consideration.
E := “Facts cited as evidence of staged burglary”. P(A|B&C&D&E) = approximately P(A|B&C&D). (Likelihood ratios involved are close to unity; certainly small relative to P(~A)/P(A).)
F := “bra clasp and knife”. P(A|B&C&D&E&F) = possibly as much as an order of magnitude higher than P(A|B&C&D). (Explaining results is a minor puzzle.)
G := “all other information”. P(A|B&C&D&E&F&G) = approximately P(A|B&C&D&E&F). (Other evidence weak; slightly inculpatory facts canceled out by slightly exculpatory facts.)
Thanks, that’s very helpful. Perhaps you could copy this to the main debate branch, so Rolf would see it and possibly respond in a similar fashion? Also, to seek a bit more clarification, what is your estimate of P(A|B&C&D) / P(A|B&C)?
The point is that the calculation you gave is missing too many steps to be useful for someone just coming to the discussion.
To take another example, does your prior take into account the gender of the suspect? (Females commit far fewer murders than males.) Or is that also screened off by some other evidence?
I suspect that a lot of the details you’re wondering about will quickly emerge in the discussion with Rolf, since (1) our opinions are widely separated, which seems to imply very different-looking calculations, with substantial inferential gaps to be bridged, and (2) that discussion is just getting started.
That said, I’m not sure which missing steps you consider the most important. A very short case summary from my point of view would be something like “student killed by burglar; housemate and boyfriend blamed before burglar discovered; after catching burglar, police filter evidence to fit three-person theory instead of dropping initial idea.” Is that helpful at all?
A reference class that gives an upper bound for my prior would be “intelligent 20-year-old female college student with no criminal history commits murder”.
Wikipedia says
Which gives no more than 0.000013 probability that Knox is a murderer if all we know is that she lives in Italy. I guess “intelligent 20-year-old female college student with no criminal history” is less likely to commit murder than average, so I’m still confused how you got “between 0.0001 and 0.001″.
Well, the answer I suppose is that I wasn’t taking the country into account.
However, if you agree that “between 0.0001 and 0.001” is an upper bound, that surely suffices! The important kind of confusion would be where you think my prior is too low, rather than too high.
I was trying to understand what evidence has been taken into account in your prior (i.e., is there some other information that might be considered Bayesian evidence against Knox, but which is already in your prior), so that I can understand what other evidence you consider “negligible”. I think at this point that confusion has been resolved.
I still wonder why the two sides don’t each post a more detailed Bayesian calculation. Let’s say A=”Knox killed Kercher”, B=”Kercher has been killed and Knoxed lived in Italy and is an intelligent 20-year-old female college student with no criminal history”, C=”evidence against Guede”, D=”Knox and Kercher were roommates”, E=”evidence of a staged burglary”, F=”bra and clasp”, G=”all other information about the case”. What are
P(A|B)
P(A|B&C)
P(A|B&C&D)
P(A|B&C&D&E)
P(A|B&C&D&E&F)
P(A|B&C&D&E&F&G)
(Or some other set of evidence and order of evaluation that might be more appropriate.) Wouldn’t that help to quickly pinpoint where your disagreements are?
I’ll redefine slightly:
A := “Knox killed Kercher, given background info about both, but not the fact of their acquaintance”. P(A) = tiny.
B := “Kercher killed”. P(A|B) = approximately P(A). (We are not yet given that they were roommates.)
C := “evidence against Guede”. P(A|B&C) = approximately P(A). (No significant connection between Guede and Knox.)
D := “Knox and Kercher were roommates”. P(A|B&C&D) = slightly higher than P(A), but still well below the threshold of consideration.
E := “Facts cited as evidence of staged burglary”. P(A|B&C&D&E) = approximately P(A|B&C&D). (Likelihood ratios involved are close to unity; certainly small relative to P(~A)/P(A).)
F := “bra clasp and knife”. P(A|B&C&D&E&F) = possibly as much as an order of magnitude higher than P(A|B&C&D). (Explaining results is a minor puzzle.)
G := “all other information”. P(A|B&C&D&E&F&G) = approximately P(A|B&C&D&E&F). (Other evidence weak; slightly inculpatory facts canceled out by slightly exculpatory facts.)
Thanks, that’s very helpful. Perhaps you could copy this to the main debate branch, so Rolf would see it and possibly respond in a similar fashion? Also, to seek a bit more clarification, what is your estimate of P(A|B&C&D) / P(A|B&C)?