Updating equations for the error rates
Consider a site and a read . The joint probability of the true nucleotide in the read, , at the site being and the observed nucleotide at the site is:
Using Bayes Theorem, the probability of the true nucleotide in the read, , at the site being , given that we observe is:
Inserting 29.15 in 29.16 we get:
The equation 29.17 gives us the probabilities for a given read, , and site, , given the observed nucleotide , that the true nucleotide is , , given our current values for the frequency (inserted for ) and error rates. Since we know the sequenced nucleotide in each read at each site, we can get new updated values for the error rate of producing an nucleotide when the true nucleotide is , , for by summing the probabilities of the true nucleotide being for all reads across all sites for which the sequenced nucleotide is , and dividing by the sum of all probabilities of the true nucleotide being a across all reads and all sites: