Updating equations for the multinomial model frequency parameters
Consider a site, , and let be the nucleotide observed in read at this site, . For each of the multinomial models that may explain the data at the site we have a number of frequency parameters. For simplicity, we consider the model which states that there are two alleles present at the site, the reference allele, , and another allele , and let be the frequency parameter for the non-reference allele (hence the frequency of the reference allele, , is ). Models with more alleles are treated in a similar manner.
We want to estimate the parameter for the frequency of the allele at the site , , by the fraction of true nucleotides that are at this site,given the observed data:
To calculate this we use Bayes Theorem on the numerator:
Inserting our current values for the frequency parameter under the model, and the error rates and , in 29.12, and further inserting the obtained values in 29.11 gives us updated values for the frequency parameter .