Mapping the drivers of within-host pathogen evolution using massive data sets

Palmer, Duncan S.; Turner, Isaac; Fidler, Sarah; Frater, John; Goedhals, Dominique; Goulder, Philip; Huang, Kuan-Hsiang Gary; Oxenius, Annette; Phillips, Rodney; Shapiro, Roger; Vuuren, Cloete van; McLean, Angela R.; McVean, Gil

The right hand side of (4) is known as the product of approximate conditionals (PAC) likelihood as each conditional likelihood P (D k+1 |D 1 , D 2 , . . . , D k , Θ) is approximated using the functionπ. Li and Stephens [1] use a hidden Markov model (HMM) forπ that captures a collection of desirable properties of the true likelihood function, but is computationally tractable, because that the (k + 1) th sequence is an imperfect mosaic of the first k sequences due to mutation and recombination. For the (k + 1) th sequence, the hidden states are the first k sequences. We denote X 1 , X 2 , . . . , X m as the hidden Markov chain that emits the sequence D k+1 , where X j is the sequence being copied from (SCF) at position j and m is the length of the sequence. Between each site, the SCF switches to a new sequence l with probability q x,i (j) (which may depend on the current SCF; x, the new SCF; i, and the current site; j). Given a hidden state at position j, the probability of observing the codon in the observed sequence D k+1 is generated by a model of codon substitution. To generateπ(D k+1 |D 1 , D 2 , . . . , D k , Θ) we integrate over all possible paths through sequences D 1 , D 2 , . . . , D k that can generate the sequence D k+1 through a combination of jumping between these k sequences (recombination) and error in copying (mutation), as illustrated in Figure 1b .