Appendix
The following section includes a more detailed description of the notation used in the paper. First, we assume locus i, (i = 1, … , I) has A
i
possible alleles, and let p
i
be a vector specifying the probabilities of a haplotype’s alleles at locus i given the haplotype’s alleles at lower indexes. We let r
2, … , r
I
denote the recombination rates between the loci, which are assumed known. For a locus i, let t be a transmission, specifying a start allele in the parent, a resulting allele in the child, and whether the parent is a mother or a father. We then denote with m
i
(t) the probability that the child obtains the resulting allele, given that the parent has the start allele. This function specifies the mutation model at locus i. The parameters of our model are p = (p
1, … , p
I
), r = (r
2, … , r
I
), and m = (m
1, … , m
I
).
If parents’ alleles follow the population frequencies, the probabilities for a child to have various alleles are not given by the population frequencies, unless the process represented by the mutation model happens to have the population frequencies as stationary distribution. This means that adding the untyped father or mother to a person in the pedigree may change the probability results we are computing. To avoid this nuisance, we recommend that all untyped founders with only one child in the pedigree are (recursively) removed prior to computations. In our pedigree, a person may have specified no parents, only a mother, only a father, or both parents. Founders are those who have no parents in the pedigree. We also assume the pedigree does not contain untyped children with no descendants as such children cannot affect the result.
Our observed data is divided into data s for S typed founders and data d for M typed non-founders: Let s
i
j
for i = 1, … , I, j = 1, … , S denote the observed allele or alleles of typed founder j at locus i. For males and X- chromosomal data, s
i
j
specifies only one allele, otherwise s
i
j
specifies the two observed alleles in no particular order. For the typed non-founders, let d
i
j
specify the similar data. We write s
i
= (s
i1, … , s
i
S
), s = (s
1, … , s
I
), d
i
= (d
i1, … , d
i
M
), and d = (d
1, … , d
I
).
We also need a number of ancillary variables: The inheritance pattern at locus i can be described as a vector v
i
of length N, with one component for each parent-child relationship in the pedigree when the locus is autosomal, and one for each mother-child relationship for X- chromosomal loci. Each component is 0 or 1 depending on whether the paternal or maternal allele is inherited, we write v = (v
1, … , v
I
). We also need to describe the founder alleles of the pedigree: These are maternal or paternal alleles whose relevant parent is not in the pedigree. First, there are founder alleles belonging to typed founders: Let g
i
j
be the allele or alleles of typed founder j at locus i listed with the paternal allele first. Write g
i
= (g
i1, … , g
i
S
) and g = (g
1, … , g
I
). For the remaining F founder alleles, let f
i
j
denote the j
′
t
h founder allele at locus i. Finally, we write f
i
= (f
i1, … , f
i
F
) and f = (f
1, … , f
I
).