Let and denote a pair of observed (ordered) parental and child haplotypes respectively. Here denotes the th parental haplotype at the sites across a chromosome. The same notation is used for the th child haplotype . There are 4 possible patterns of gene flow between the parent and child. The true pattern of gene flow will remain constant over long stretches of a chromosome due to the low rate of recombination in any given meiosis. We use to denote the pattern of gene flow at the th locus, where means that the parents th haplotype and the child's th haplotype are identical by decent (IBD). Here , so there are just 4 possible inheritance patterns, which we denote . The true inheritance states are unobserved across each chromosome and we wish to infer them from our imperfect observations of the parental and child haplotypes and . The intuition behind our approach is that true recombination events and SEs will cause changes to the pattern of gene flow as we move along a chromosome. Since we expect just a few true
