= Jacquard identity coefficients
{c}
{title2=$\Delta_1,\ldots,\Delta_9$}
For two individuals carrying labelled copies $(a,b)$ and $(c,d)$, condensed identity coefficients give the <probabilities> of nine <identity by descent> patterns, identifying states that differ only by exchanging the two homologues within either individual. They retain within-individual <autozygosity>, which three ordinary outbred sharing coefficients omit. One convention is
$$
\begin{array}{c|l}
1&abcd\\2&ab\mid cd\\3&abc\mid d\text{ or }abd\mid c\\4&ab\mid c\mid d\\5&acd\mid b\text{ or }bcd\mid a\\6&cd\mid a\mid b\\7&ac\mid bd\text{ or }ad\mid bc\\8&\text{one cross-individual pair, two singletons}\\9&a\mid b\mid c\mid d
\end{array}
$$
Each block consists of copies identical by descent, and different blocks have distinct ancestral origins. For siblings whose parents are otherwise outbred <first cousins>, these coefficients are $(1,0,2,1,2,1,15,30,12)/64$. This follows by conditioning on whether the parents share one ancestral copy and enumerating the four independent <Mendelian segregation> choices to their two children. The <kinship coefficient> is $\Delta_1+(\Delta_3+\Delta_5+\Delta_7)/2+\Delta_8/4$, here $9/32$. https://pmc.ncbi.nlm.nih.gov/articles/PMC2585748/[Weir's discussion of identity states] uses this convention.
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