Currently, I assume all pairs of observations are independent so, for example in one deme, the log likelihood becomes:
log(lambda) * sum_{i=1}^{n'} x_i - n'lambda
where n' = n(n-1)/2 and n is the number of samples.
The problem is that the likelihood thinks there are n' data points when there might only be n points. In the one deme example, the idea is to divide by (n-1)/2 to adjust for the sample size. Another way to view it is that I am making the likelihood less pointy.
After some algebra, this weighted composite likelihood simplifies to:
one deme case: n (log(lambda) xbar - lambda)
two deme case: (m+n) (log(lambda) xbar - lambda) where m is the number of individuals in deme 1 and n is the number of individuals in deme 2.
You can see that the mle estimate is still xbar (the mean IBD sharing).
So, to test whether this weighting works, I will simulate a barrier under macs and investigate the performance of this weighted composite likelihood vs the independent pairwise likelihood. Also, I should simulate a barrier under unequal sample size.
Currently, I assume all pairs of observations are independent so, for example in one deme, the log likelihood becomes:
log(lambda) * sum_{i=1}^{n'} x_i - n'lambda
where n' = n(n-1)/2 and n is the number of samples.
The problem is that the likelihood thinks there are n' data points when there might only be n points. In the one deme example, the idea is to divide by (n-1)/2 to adjust for the sample size. Another way to view it is that I am making the likelihood less pointy.
After some algebra, this weighted composite likelihood simplifies to:
one deme case: n (log(lambda) xbar - lambda) two deme case: (m+n) (log(lambda) xbar - lambda) where m is the number of individuals in deme 1 and n is the number of individuals in deme 2.
You can see that the mle estimate is still xbar (the mean IBD sharing).
So, to test whether this weighting works, I will simulate a barrier under macs and investigate the performance of this weighted composite likelihood vs the independent pairwise likelihood. Also, I should simulate a barrier under unequal sample size.