Let \(\{ y_i \}_{i=1}^n\) denote the coverage at position \(i\) for \(n\) loci across the chromosome/genome.
Let \(\{ s_i \}_{i=1}^n\) denote the segment number for each of those locations where \(s_i \in \{ 1, \dots, S \}\) and \(S\) is the total number of segments.
\begin{align} p(y_i | s_i = j, w, \mu, \sigma^2 ) = (1-\rho_{j}) \underbrace{\sum_{c=1}^{C} w_{0,c} f( y_i; \mu_{0,c}, \sigma_{0,c}^2 )}_{\text{Common}} + \rho_{j} \underbrace{ \sum_{k=1}^{K} w_{j,k} f( y_i; \mu_{j,k}, \sigma_{j,k}^2 ) }_{\text{Segment-specific}} \end{align}where \(f\) is the density function for the Normal distribution, \(\sum_{k=1}^K w_{j,k} = 1\) and \(\sum_{c=1}^{C} w_{0,c} = 1\).
For each data point \(y_i\), \(\psi_{i}\) gives the probability it is in a common (rather than segment specific) component.
\begin{align} \psi_{i} = \frac{ (1-\rho_{j}) \sum_{c=1}^{C} w_{0,c} f( y_i; \mu_{0,c}, \sigma_{0,c}^2 ) } { (1-\rho_{j}) \sum_{c=1}^{C} w_{0,c} f( y_i; \mu_{0,c}, \sigma_{0,c}^2 ) + \rho_{j} \sum_{k=1}^{K} w_{j,k} f( y_i; \mu_{j,k}, \sigma_{j,k}^2 ) } \end{align}For each data point \(y_i\), \(\phi_{i,c}\) gives the probability it is in component \(c\) given it’s in a common component. \(\nu_{i,k}\) gives the equivalent for segment specific components.
\[ \phi_{i,c} = \frac{ w_{0,c} f(y_i;\mu_{0, c}, \sigma_{0,c}^2 ) } { \sum_{c=1}^{C} w_{0,c} f(y_i;\mu_{0,c}, \sigma_{0,c}^2 ) } , \]
\[ \nu_{i,k} = \frac{ w_{s_i,k} f(y_i;\mu_{s_i,k}, \sigma_{s_i,k}^2 ) } { \sum_{k=1}^{K} w_{s_i,k} f(y_i;\mu_{s_i,k}, \sigma_{s_i,k}^2 ) } \]
M update for \(\rho\), a global common vs segment specific weighting:
\[ \rho_{j} = 1 - \frac{ \sum_{i : s_i = j} \psi_i }{ \sum_{i : s_i = j} 1 } \]
M updates for common component parameters:
\[ w_{0,c} = \frac{ \sum_{i=1}^n \psi_i \phi_{i,c} } { \sum_{i=1}^n \psi_i } \]
\[ \mu_{0,c} = \frac{ \sum_{i=1}^n \psi_i \phi_{i,c} y_i } { \sum_{i=1}^n \psi_i \phi_{i,c} } \]
\[ \sigma_{0,c}^2 = \frac{ \sum_{i=1}^n \psi_i \phi_{i,c} ( y_i - \mu_{0,c} )^2 } { \sum_{i=1}^n \psi_i \phi_{i,c} } \]
M updates for segment specific component parameters:
\[ w_{j,k} = \frac{ \sum_{i : s_i = j } (1-\psi_i) \nu_{i,k} } { \sum_{i : s_i = j} (1-\psi_i) } \]
\[ \mu_{j,k} = \frac{ \sum_{i : s_i = j } (1-\psi_i) \nu_{i,k} y_i } { \sum_{i : s_i = j} (1-\psi_i) \nu_{i,k} } \]
\[ \sigma_{j,k}^2 = \frac{ \sum_{i : s_i = j } (1-\psi_i) \nu_{i,k} ( y_i - \mu_{j,k})^2 } { \sum_{i : s_i = j} (1-\psi_i) \nu_{i,k} } \]