Rigorous definition of the kinship coefficient and proof of a recursion thereof

Question

I am reading Section 5.2, Kinship and Inbreeding Coefficients, of Kenneth Lange, Mathematical and Statistical Methods for Genetic Analysis. There the kinship coefficient $\Phi_{i,j}$ is defined for two relatives $i$ and $j$ as the probability that a gene selected randomly from $i$ and a gene selected randomly from the same autosomal locus of j are identical by descent. Then the book states that suppose $k$ and $l$ are the parents of $i$, $$\Phi_{i,j}=\frac12(\Phi_{k,j}+\Phi_{l,j}) \tag1$$ with some exceptions.

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Equation (1) seems intuitive. Yet it is slippery for a proof especially considering the exceptions. The root cause is that the definition of $\Phi_{i,j}$ is vague albeit seemingly plausible. I am seeking a mathematically rigorous definition of $\Phi_{i,j}$ and a proof for Equation (1) together with the condition under which it applies.

Answer 1

The definition as set by Kenneth Lange's book is indeed quite vague and thus unrigorous. It is a probability not conditioned on observing an allele $a$ at a locus of $i$ but conditioned on the following where we focus on one particular locus. It is rigorously defined as follows. Then we will give the correct formulation of the correct recursive equation akin to Equation $(1)$ which is in general wrong.

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A graph of lineage contains nodes, which we call persons, of a particular locus. Each node, or person, contains exactly two genes. We draw one and only one directed edge (parental relationship) from a gene of a parent person to the gene of a child person if the latter is inherited from the former. This is a directed acyclic graph (DAG). Each gene can be connected by no more than $1$ edge directed to (as opposed to away from) it. The two genes in one person cannot be connected by edges. Two persons are said to be connected if there is an edge pointing from a gene of one node to a gene of the other node. Two persons cannot be connected by more than one edge. For a pair of connected persons and their directed edge between them, the edge is equally probable to connect to one gene as the other gene of a person, i.e. the probability is $\frac12$. A pair of genes (persons) are said to be connected if there exists a undirected (simple) path from one gene (person) to the other. Two connected genes are called identical by descent.

In this setting, the DAG with respect to the persons is given and deterministic whilst any directed edge in the DAG assigned to two persons are random with independent probability $\frac12$ with respect to the genes within each person.

For a pair of persons on a lineage DAG that is connected, the probability of one gene equally probably chosen amongst the two genes of one person being connected, or identical by descent, to one gene equally probably chosen amongst the two genes of the other person is the coefficient of kinship.

As an additional assumption, the genes that are connected have to be the same and thus assume the same allele.

Definition: Suppose a lineage DAG is given. Let the kinship coefficient between person labeled $1$ and person labeled $2$ be $\Phi(1,2)$. Denote the set of parents and children of person $1$ each of which is on an undirected (simple) path between $1$ and $2$ as $S_2(1)$.

Lamma: Denote the set of paths through an ordered persons $p$ in that order as $P(p)$. $\{P((1,i,2))|i\in S_2(1)\}$ partitions $P((1,2))$.

Proposition: $$\Phi(1,2)=\frac12\sum_{i\in S_2(1)}\Phi(i,2).$$
Note: The summation in the Proposition is over $S_2(1)$ rather than the set of all parents and children of person $1$. That fixes the condition under which Equation $(1)$ is valid.

Corollary: $$\Phi(1,2)=\sum_{p\in P((1,2))} \frac1{2^{n(p)}}$$ where $n(p)$ is the number of persons on path $p$. $n(p)=|p|+1$.