Table of contents
YA Carter and BI Nestor both have scars?
Let's be more precise. The evidence is that
YA Carter has scars and relatives report that the infant BI Nestor had scars.
More carefully, let's describe the evidence as EC & EN, where
|EC =||YA Carter has scars of a certain description S.|
|EN =||BI Nestor's family describes scars of a certain description S' (similar to but certainly not identical description to S)|
X = Pr(EC & EN | Carter is Nestor)
Y = Pr(EC & EN | Carter is not Nestor).
|... doesn't depend on the relationship||The way people describe Carter is not biased by his true history (especially if unknown).|
Note: EC doesn't depend on the relationship, so Pr(EC | Carter is Nestor) = Pr(EC | Carter is not Nestor) = Pr(EC).
Consider X. There are two ways to apply the identity Pr(F & G) = Pr(F)Pr(G|F). In this case I choose an adult-centric formulation by letting EC play the role of F:
So LR=Pr(EN | N*)
/ Pr(EN | ~N*), where
|Pr(EN | H1)||= The probability that relatives of a person with scars S would report childhood scars S'|
|Pr(EN | H0)||= The probability that relatives of some random infant would recall childhood scars S'.|
|Evidence EN' =||Relatives remember and report scars S' for the child Nestor after being given a suggestion in the form of information or a picture of Carter.|
We have some survey data based on 698 missing children:
|... are represented by some part of the sample of 698||"Represented" doesn't mean Carter would be among the 698 in the sample, but only that, as the sample represents the 30,000 or so missing children, the missing children with Carter's age and time of disappearance are proportionally represented in it.|
6 ≤ LR ≤ 36.
Considering the total evidence LRT to be composed of two factors, scientific (meaning DNA) and anecdotal (everything else), we have
|LRT||= LRDNA × LRother|
|= 11,700 × LRother|
|≥ 30,000,000, so we need|
|LRother||≥ 30,000,000/11,700 = 2564.|
|LR shortfall thinking||requisite prior thinking|
|LRother ≥ 2564 means
that the DNA evidence leaves us with a LR shortfall of 2564.
Can we do it? Suppose we figure
LRscar ≥ 2564 / (2 × 12) = 107.
We could say that the LR shortfall before consideration of the scar is 107.
Is LRscar ≥ 107? That's believable, but it's not obvious! Might depend on the details.
LRsex — why it is 2evidence E = gender of BI is same as gender of YA.
|The "requisite prior" thinking means to consider the "other" evidence before the DNA,
and wrap it into the prior odds.