content_image/295736.jpg content_image/295735.jpg
between analytical mode and iteratively computed mode. 

// Initialization Set the initial values γ$_{1}$, γ$_{2}$, N , m$_{0}$ , g . // Main loop while $^{∑}$| m new i − m$_{i}$ | > 10 − 5 do m new i = max[ π ( m$_{i}$ | t$_{i}$, g$_{i}$, N, φ , α , γ$_{1}$, γ$_{2}$ ] = ( g$_{i}$ + α ) / ( N ) g new i = max[ π ( g$_{i}$ | T , m , N, φ , α , γ$_{1}$, γ$_{2}$ ] = t$_{i}$ + floor( Nm$_{i}$ (1 − φ$_{i}$ )) N new = max[ π ( N | T , g , m , φ , α , γ$_{1}$, γ$_{2}$ ] = ( $^{∑}$g$_{i}$ + γ$_{1}$ ) / (1 + γ$_{2}$ ) end 

## 4.2 A Dirichlet-Multinomial-Binomial Approach. 

A second approach is both more computationally efficient and arguably more directly compa- rable to the sampling mechanism inherent in SAGE. For convenience, we begin by assuming that the total number of mRNA in the sample is N ∼ P ois ( λ ). Given this mRNA popula- tion size, the vector g of counts of each category of mRNA prior to tag formation follows a multinomial distribution, 

<img src='content_image/295736.jpg'>

whose probabilities are the m$_{i}$ and are assumed to follow a Dirichlet distribution with pa- rameter vector α . 

The joint posterior distribution can now be written as, 

<img src='content_image/295735.jpg'>

