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

89
which can be shown to have higher likelihood than θ t .
The details of these calculations depend on the emission model. We consider two specific
models for the emission distribution (B.1). In the first model, η = (λ 1 , . . .,λ K , σ), which
defines the emission distributions
X i |Π i = k ∼ N B (λ k , σ) (B.5)
In words, the emission distribution in each state is negative binomial with a different mean
and a common size parameter. The second model attempts to ensure that copy number
changes are limited to simple fractions (e.g. 2:3, 1:2, 1:1, 2:1, 3:2) by fixing them relative to
each other. Specifically, η = (λ, σ) and
X i |Π i = k ∼ N B (λα k , σ) (B.6)
where α 1 , . . .,α K are fixed known constants. Theorem 1 gives the Baum-Welch updates for
these two models in terms of the quantities
A k,l =
B k =
N k,b =
∑L−1
i=1
∑L−1
i=1
∑L−1
i=1
I {Πi =k,Π i+1 =l}
I {Πi =k}
I {Πi =k,x i =b}
and
Ā k,l (θ) = E (A k,l |x, θ)
¯B k (θ) = E (B k |x, θ)
¯N k,b (θ) = E (N k,b |x, θ)
The proof of the theorem is long but straightforward, and will not be given here.