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Questions about calculating the genotype likelihoods

In this website, https://software.broadinstitute.org/gatk/documentation/article.php?id=4442, you showed the formula used to calculate PL.

I can understand most of the formulas used here. But I can't understand the change on the formula when you are trying to implement G=H1H2 to P(D|G). I tried a lot of times and I cannot finish the math inference on my own. I think the formula you used to calculate P(D|G) should also be available to be generated by pure math deduction.

Therefore, if convenient, would you please show me the process of the math deduction of the formula to prove that P(D|G)=P(D|H1)/2 + P(D|H2)/2 (given a single read sequence).

Thank you!


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