Table 1. List of symbols and their brief descriptions.

Symbol	Description
$n_F$	Number of founders
N	Number of offspring
T	Number of markers (loci)
$h_t^F$	Hidden founder haplotype at locus t
${\mathit{h}}^F$	Hidden founder haplotype matrix ${\mathit{h}}^F={\left\{h_t^F\right\}}_{t=\mathrm{1...}T}$
$x_{ti}$	Hidden ancestral origins at locus t in offspring i
$x_{ti}^m,x_{ti}^p$	$x_{ti}=\left(x_{ti}^m,x_{ti}^p\right)$ on maternally (m) or paternally (p) derived chromosome
$d_{ti}$	Genotype at locus t in offspring i that is completely determined by $x_{ti}$ and $h_t^F$
$z_{ti}$	Hidden true genotype at locus t in offspring i
$y_{ti}$	Observed genotype at locus t in offspring i
${\mathit{y}}^O$	Observed offspring genotype matrix ${\mathit{y}}^O={\left\{y_{ti}\right\}}_{t=\mathrm{1...}T,i=\mathrm{1...}N}$
$y_t^F$	Observed genotypes for all founders at locus t
${\mathit{y}}^F$	Observed founder genotype matrix ${\mathit{y}}^F={\left\{y_t^F\right\}}_{t=\mathrm{1...}T}$
$1U,2U,UU$	Genotypes containing uncertain allele U
$r_1,r_2$	Number of reads for alleles 1 or 2
${\mathit{ϵ}}_O$	Allelic error probability for offspring, independent of read depths
${\mathit{ϵ}}_F$	Allelic error probability for founders, independent of read depths
$phred$	Phred quality score
ε	Sequencing error probability $\epsilon ={10}^{-phred/10}$
$\pi \left(x_{1i}\right)$	Prior probability of $x_{1i}$ at locus 1 in offspring i
$P\left(x_{ti}|x_{t-1,i}\right)$	Prior transition probability from $x_{t-1,i}$ to $x_{ti}$
$l_{ti}$	$l_{ti}=P\left(y_{ti}|h_t^F,x_{ti},{\mathit{ϵ}}_O,{\mathit{ϵ}}_F,\epsilon \right)$ likelihood at locus t in offspring i
$\alpha \left(x_{ti}|h_t^F\right)$	Posterior probability of $x_{ti}$ conditional on $h_t^F$ and genotypic data from loci 1 to t
$\stackrel{\sim }{\alpha }\left(x_{ti}|h_t^F\right)$	Unnormalized conditional posterior probability of $x_{ti}$
$\gamma \left(h_t^F\right)$	Posterior probability of $h_t^F$ conditional on genotypic data from loci 1 to t
${\widehat{h}}_t^F,{\widehat{x}}_{ti},{\widehat{z}}_{ti}$	Hats denote maximum likelihood estimates
$P_{\text{call}}$	Single genotype call if probability of most probable genotype is greater than threshold $P_{\text{call}}$
$P_{\text{impute}}$	Impute if probability of most probable genotype is greater than threshold $P_{\text{impute}}$
$P_{\text{detect}}$	Correct if probability of most probable genotype is greater than threshold $P_{\text{detect}}$