H is the set of all possible Diplotypes that are consistent with genotype data

Question

I am new to Biology.SX. I have a statistics background, and have almost zero knowledge in genetics. I am trying to understand some things related to genetics in a certain paper of biostatistics.

1. Haplotype data were simulated using the haplotype patterns and frequencies (shown below) for 5-SNPs along a dabetes susceptibility region on chromosome 22, reported in FUSION study.

I know that haplotypes can be represented as binary sequences. I wonder why all the possible $2^5$ are not present here. (?)

The paper also says that

2. Let $G=(g_1,\ldots,g_M)$ denote the unphased genotype data for the $M$ loci. $\mathcal{H}_G$ denote the set of all possible diplotypes that are consistent with the genotype data $G$.

If a subject carries at most one copy of the causal haplotype '01100', it belongs to dominant genetic model, if it carries two copies of this haplotype, it belongs to recessive model.

The set $\mathcal{H}_G$ is not clear to me. While computing the likelihood, I need to know $\mathcal{H}_G$. Any help or suggestions?

Answer 1

Disclaimer: Like the OP, I know very little about genetics and I suppose other people in the site can give better answers than me. Anyway, since the question has been unanswered for months I'm posting my answer. Hopefully someone will improve it.

Just beware that "possible" in this context doesn't mean all haplotypes that we can imagine. As the paper says, "$\mathcal{H}_G$ denote the set of all possible diplotypes that are consistent with the genotype data $G$" and that is an additional restriction that allows just some diplotypes - the seven ones listed, not the 32 we can imagine.

Furthermore, please notice that sum of frequencies in the table equals 1, therefore there can be no more haplotypes in the set.

However, it would help if you linked the source where you got those data and quotes.