

For example, I’ve been asked why not +2 in the denominator, i.e. a formula of
(x_{i}+1)/(N+2).I partially understand the sense of that. If the +1 in the recommendation is to condition the probability on the observed instance of the paternal allele in the child, then wouldn’t it be more accurate to condition also on the noninstance of that allele as the child’s other allele?
So I agree that the N+2 formula isn’t illogical. However it seems to me an unnecessary complication for negligible benefit.
Therefore where is one to stop? In a spirit of practicality, I am inclined to stop at the earliest step as long as it is adequately accurate. Otherwise, we can spend endless hours on trivialities.
(x_{q}+1)/(N+3)?
(x_{q}+2)/(N+3).
Don’t give up! There must be a probabilistic answer. Let’s put
Pr(P  Mother, Child types) = (x_{p}+j) / (N+3)with
Pr(Q  Mother, Child types) = (x_{q}+k) / (N+3)
j+k=3.Maybe it is reasonable to take j and k to be proportional to x_{p} and x_{q}, giving rise to formulas like
Pr(Q  Mother, Child types) = x_{q}[1+3/(x_{q}+x_{p})] / (N+3).Is that attractive?
The simpler formula x/N is too simple; it’s biased against the alleged father and the bias is significant for small x. (The formula (x+1)/N may seem simpler, but I don’t like it for two reasons:
In the pursuit of greater accuracy you generally run into endless complication with zero or negligible benefit. Only in the homozygous child QQ case it might be justifed to accept the complication of +2, rather than +1. It does correct a situation where the recommendation is anticonservative. However, I am reluctant to recommend it because: