Download Annotation of genomics data using bidirectional hidden Markov

µi + µī =
gi
gi
+λ=
+2
πi
πi
(15)
Plug (15) into Equation (11) to obtain
zij
= µi + µj̄
ρij
=
(µi + µī ) + (µj + µj̄ ) − (µj + µī )
=
gj
zji
gk
+2+
+2−
πi
πj
ρji
(16)
Use τij = zij ρ−1
ij
τij + τji =
gi
gj
+
+4
πi
πj
(17)
for τij and hence determine ρij . We guess an approximate solution
τij =
gi + γ0 (i) + γT (ī)
πi
(18)
ρij =
πi zij
gi + γ0 (i) + γT (ī)
(19)
or
Note, that by choosing this specific solution, we are able to guarantee that the
updated parameters satisfy all constraints of a bdHMM (see below). However,
we cannot guarantee that the updated parameter set will have a marginal likelihood larger or equal to the previous parameter set θold .
With this choice for ρij , condition (5) is fulfilled:
∑
zij
=
j
∑
(Zij + Zj̄ ī ) =
T ∑
∑
t=1
j
=
T
∑
(ζt (i, j) + ζt (j̄, ī))
j
(γt (i) + γt (ī))
t=1
= γ0 (i) +
T
−1
∑
(γt (i) + γt (ī)) + γT (ī)
t=1
= γ0 (i) + Gi + Gī + γT (ī)
= gi + γ0 (i) + γT (ī)
(20)
And consequently,
∑
j
ρij =
∑
πi
zij = πi
gi + γ0 (i) + γT (ī)
4
(21)