## 4.1 Gibbs Sampling based on a Dirichlet-Poisson-Binomial Model. 

Mechanically, the SAGE technique proceeds as discussed in Section 2 where the pool of mRNA from the cells are first converted to cDNA, tagged and then amplified. In the model discussed below, inspired by Cassella and George, we assume that a fixed population of mRNA’s exists of size N which we will suppose is random within the cells. The total size N is determined by a gamma distribution which is rounded off to the nearest integer. The choice of gamma here is convenient due to its role as a conjugate prior. Given the population of size N , the relative proportions m$_{i}$ of different categories of mRNA are unknown and may be modeled as a Dirichlet distribution with prior α = ( α$_{1}$, . . . , α$_{k}$ ) where the α$_{i}$ will often be chosen to be identical, i.e. α$_{i}$ = c ∀ i and some constant c . 

Because cells contain mRNA transcripts from thousands of different genes, the probability of seeing any particular gene is low. It is therefore logical to assume, given N and m$_{i}$ , that the actual number of mRNA’s of a certain type g$_{i}$, i = 1 , . . . k , extracted from the group of cells, satisfies a Poisson law with mean µ = N · m$_{i}$ . Finally, the restriction enzyme process generates a tag count t$_{i}$ for a particular complimentary DNA strand in a binomial fashion based upon the tag formation probability φ$_{i}$ . Hence, the hierarchical data generating mechanism follows, T$_{i}$ ∼ Binomial ( g$_{i}$, φ$_{i}$ ), g$_{i}$ ∼ P oisson ( N ∗ m$_{i}$ ), m ∼ Dirichlet ( α$_{1}$, . . . , α$_{k}$ ), N ∼ ceiling ( Gamma ( γ$_{1}$, γ$_{2}$ )). We refer to this model as the Dirichlet-Poisson-Binomial (DPB) model. A first weakness of this model is that the sum of all counts $^{∑}$g$_{i}$ may not add up to the total counts N . However, this approach allows us to infer the total population size N . In constructing this hierarchical model, we have essentially provided a multivariate extension of the model investigated in Thygesen & Zwinderman (2006) while integrating the 

