Derivation of EM algorithm - PLOS

Derivation of EM algorithm The complete log-likelihood including missing data {zi } for the proposed model is I X J X K X

I(zi,j = k)

L X

i=1 j=1 k=1

 log fk,l,xi,j,l + log qi,k .

l=1

Here, we introduce the variable for conditional probability for zi,j given the parameters and the mutation features xi,j ,
θi,k,m = Pr zi,j = k xi,j = m, {fk,l }, {qi } Note that this conditional probability just depends on the value of mutation feature m = (m1 , · · · , mL ), not on the index j. Then, the expected complete log-likelihood augmented by Lagrange multipliers is calculated as I X X i=1 m

gi,m

K X

θi,k,m

k=1

L X



log fk,l,ml + log qi,k +

l=1

K X L X k=1 l=1

τk,l 1 −

Ml X

I K X
X
fk,l,p + ρi 1 − qi,k .

p=1

i=1

k=1

Differentiating it leads to following stationary equations: I X X

gi,m θi,k,m − τk,l fk,l,p = 0, (p = 1, · · · , Ml , k = 1, · · · , K, l = 1, · · · , L).,

i=1 m:ml =p

X

gi,m θi,k,m − ρi qi,k = 0, (k = 1, · · · , K, i = 1, · · · , I).

m

Then, by eliminating Lagrange multipliers, updating rules can be obtained.

1