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komponisto comments on Inherited Improbabilities: Transferring the Burden of Proof - Less Wrong
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Ahhh, you make so much more sense when you phrase it this way!
"other evidence that Knox and Sollecito killed Kercher could raise the probability of their having faked the burglary"
But my point is, this is backwards. It only works if you assume with near-100% certainty that faking the burglary and being the murderer are correlated. Otherwise "faked the burglary" IS simply evidence that Knox is the murderer.
If we prove that Knox killed Kercher, it proves that any 100% correlation is true. It does NOT prove any less-than-100% correlation. It's even entirely possible for a correlation to be one-directional (A implies B, but B does not imply A).
Thus, Knox killed Kercher is only proof of a faked burglary if you already assume the correlation is proven and two-directional.
Theorem: If A is evidence of B, then B is also evidence of A.
Proof: To say that A is evidence of B means that P(A|B) > P(A|~B), or in other words that P(A&B)/P(B) > P(A&~B)/P(~B), which we may write as P(A&B)/P(B) > (P(A)-P(A&B))/(1-P(B)). Algebraic manipulation turns this into P(A&B) > P(A)P(B), which is symmetric in A and B; hence we can undo the manipulations with the roles of A and B reversed to arrive back at P(B|A) > P(B|~A). QED.
Hence, if A implies B, then B also implies A!
Now of course, the strengths of these implications might be vastly different. But that's a separate matter.
Here, the point is that A implies B with near certainty (where A is "K&S faked burglary" and B is "K&S killed Kercher"); I'm not terribly concerned with how strongly B implies A. I don't need for B to imply A very strongly to make my point, but Massei and Cristiani would definitely need that in order to enable any charitable reading of their burglary section at all.