On the Analysis of Optical Mapping Data - University of Wisconsin ...

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Theorem 1. For model (B.5), where θ consists of ((a k,l )), λ k and σ,
Q(θ|θ t ) =
K∑
k=1
+ ∑ i
−
+
∑ i
∞∑
K∑
Ā k,l (θ) log a k,l
l=1
b=0 k=1
The updates for a and η are given by
a t+1
k,l
=
λ t+1
k
=
log Γ(x i + σ) − (L − 1) log Γ(σ)
log x i ! + σ
K∑
K∑
k=1
( ) σ
¯B k (θ) log
σ + λ k
¯N k,b (θ) b log
(
λk
σ + λ k
)
Ā k,l (θ)
∑ K
, 1 ≤ k, l ≤ K
l=1 Āk,l(θ) (B.7)
∑ ∞
b=0 b ¯N k,b (θ)
, 1 ≤ k ≤ K (B.8)
¯B k (θ)
σ t+1 has no closed form solution, but can be obtained by numerically solving a one dimensional
optimization problem after substituting (B.7) and (B.8) in Q(θ|θ t ). (B.7) and (B.8)
also hold in the limiting case, as σ → ∞, when the emission distribution can be approximated
by a Poisson distribution. For model (B.6),
Q(θ|θ t ) =
K∑
k=1
+ ∑ i
−
+
∑ i
∞∑
K∑
Ā k,l log a k,l
l=1
b=0 k=1
log Γ(x i + σ) − (L − 1) log Γ(σ)
log x i ! + σ
K∑
K∑
k=1
( )
σ
¯B k log
σ + λα k
¯N k,b b log
( λαk
σ + λα k
)
In this case neither σ t+1 nor λ t+1 have a closed form solution, and both have to be obtained
numerically. For the limiting Poisson model, λ t+1 is given by
∑
λ t+1 i
=
x i
∑ K ¯B k=1 k α k
(B.9)