132
E. A. Thompson
Even for a pair of individuals, the IBD of 0, 1, or 2 gametes at a location (Fig. 6.3)
are not the only possibilities. If the parents of an individual are related, then the
individual is inbred, and the two gametes within an individual may be IBD at some
locations. For four gametes, there are 15 possible IBD states; the left-hand columns
of Table 6.2 show the states in pictogram form. An alternative way to specify the
IBD is by specifying the partition of the four gametes into the IBD subsets, and
this specification is also given in Table 6.2. For example, in state 7 of Table 6.2 the
maternal gamete A m is IBD to both gametes B p and B m of B. The two subsets of
this IBD partition are thus {A p } alone and the three IBD gametes {A m , B p , B m }.
One advantage of specification as a partition is that it extends to any number of
individuals or set of haploid genomes.
For model-based inference of IBD from genetic data, a prior model for IBD
is required; see Sect. 6.3.4. Given a pedigree, the probabilities of Mendelian
segregation and the process of meiosis provide probabilities of each IBD partition
or state. However, in the absence of a known pedigree, an alternative approach is
necessary. One natural model is that of the Ewens Sampling Formula (ESF: Ewens
1972), which provides a one-parameter model for partitions of an exchangeable set
of gametes. The ESF probabilities are in general written in terms of a i , the number
of subsets of size i; this specification is given in the next column of Table 6.2.
For example, in state 7, with partition {{A p }, {A m , B p , B m }}, there is one subset of
size 1 and one of size 3: a 1 = a 3 = 1. Since, under the ESF model, the gametes
are exchangeable, all IBD partitions with the same set of values of a i must have
the same probability. For example, in Table 6.2, states 2, 9, and 10 have the same
probability since each has a 2 = 2, even though in state 2 the IBD is between the
two gametes within each individual and in states 9 and 10 there are two pairs of
inter-individual IBD.
For our purposes the probabilities are most easily parameterized in terms of β,
which is the pairwise probability of IBD between any two gametes. In terms of
Ewens’ classical parameter θ of genetic variation, β = 1/(1 + θ). For the case of
four gametes the relative probability of each state is also given in Table 6.2. Each
term is normalized by the column sum (1+β)(1+2β) to give the probability. Again,
in the example of state 7, there is one non-IBD factor (1 − β) and two IBD factors β
to link the other three gametes. The general formula for multiple gametes is beyond
the scope of this chapter, but the interested reader may consult Tavaré and Ewens
(1997). It can also be checked that every pair of gametes has IBD probability β. For
example, A m ≡ B p in states 1, 3, 7, 10, and 13 for a total probability
6β 3 + 2 × 2β(1 − β) + β 2 (1 − β) + β(1 − β) 2
(1 + β)(1 + 2β)
=
β(6β 2 + 5β(1 − β) + (1 − β) 2 )
(1 + β)(1 + 2β)
= β
(6.4)
E. A. Thompson
Even for a pair of individuals, the IBD of 0, 1, or 2 gametes at a location (Fig. 6.3)
are not the only possibilities. If the parents of an individual are related, then the
individual is inbred, and the two gametes within an individual may be IBD at some
locations. For four gametes, there are 15 possible IBD states; the left-hand columns
of Table 6.2 show the states in pictogram form. An alternative way to specify the
IBD is by specifying the partition of the four gametes into the IBD subsets, and
this specification is also given in Table 6.2. For example, in state 7 of Table 6.2 the
maternal gamete A m is IBD to both gametes B p and B m of B. The two subsets of
this IBD partition are thus {A p } alone and the three IBD gametes {A m , B p , B m }.
One advantage of specification as a partition is that it extends to any number of
individuals or set of haploid genomes.
For model-based inference of IBD from genetic data, a prior model for IBD
is required; see Sect. 6.3.4. Given a pedigree, the probabilities of Mendelian
segregation and the process of meiosis provide probabilities of each IBD partition
or state. However, in the absence of a known pedigree, an alternative approach is
necessary. One natural model is that of the Ewens Sampling Formula (ESF: Ewens
1972), which provides a one-parameter model for partitions of an exchangeable set
of gametes. The ESF probabilities are in general written in terms of a i , the number
of subsets of size i; this specification is given in the next column of Table 6.2.
For example, in state 7, with partition {{A p }, {A m , B p , B m }}, there is one subset of
size 1 and one of size 3: a 1 = a 3 = 1. Since, under the ESF model, the gametes
are exchangeable, all IBD partitions with the same set of values of a i must have
the same probability. For example, in Table 6.2, states 2, 9, and 10 have the same
probability since each has a 2 = 2, even though in state 2 the IBD is between the
two gametes within each individual and in states 9 and 10 there are two pairs of
inter-individual IBD.
For our purposes the probabilities are most easily parameterized in terms of β,
which is the pairwise probability of IBD between any two gametes. In terms of
Ewens’ classical parameter θ of genetic variation, β = 1/(1 + θ). For the case of
four gametes the relative probability of each state is also given in Table 6.2. Each
term is normalized by the column sum (1+β)(1+2β) to give the probability. Again,
in the example of state 7, there is one non-IBD factor (1 − β) and two IBD factors β
to link the other three gametes. The general formula for multiple gametes is beyond
the scope of this chapter, but the interested reader may consult Tavaré and Ewens
(1997). It can also be checked that every pair of gametes has IBD probability β. For
example, A m ≡ B p in states 1, 3, 7, 10, and 13 for a total probability
6β 3 + 2 × 2β(1 − β) + β 2 (1 − β) + β(1 − β) 2
(1 + β)(1 + 2β)
=
β(6β 2 + 5β(1 − β) + (1 − β) 2 )
(1 + β)(1 + 2β)
= β
(6.4)