sampling bias. Together, the elements described above give a joint posterior distribution, From this expression, we can deduce the set of full conditional distributions (Ghosh et al., 2006), A Gibbs sampler can be implemented based on this full set of conditionals; however, a second weakness of this approach is the need to update each value of m$_{i}$ independently. Because the vector m is typically on the order of thousands or tens of thousands, this is a costly computation. In order to minimize autocorrelation between samples, our experiments stored only the last of every hundred samples for analysis. Inference for means was based upon the final 1000 of these stored samples drawn after an extensive burn in period of 400,000 simulations. The two prior parameter vectors tested were α = 1 and α$_{i}$ = 1 /k ∀ i . k = 6096 is the number of genes in the sample. The distribution of population size, N , used a shape parameter γ$_{1}$ = 100 and prior scale 1 /γ$_{2}$ = 200. The mean of a gamma with these parameters is 20000 which is somewhat larger than a natural estimate of the population size, $^{∑}$$_{i}$( t$_{i}$/φ$_{i}$ ) = 16438 . 81. This was chosen intentionally to determine if the posterior would converge to a reasonable estimate of N. Experimentalists assume that there are ∼ 15 , 000 mRNA in a cell which also 15 [ m , g , N | T , α , γ$_{1}$, γ$_{2}$ ] ∝ k ∏ i =1 $^{(}$g$_{i}$ t$_{i}$$^{)}$ φ t$_{i}$ i (1 − φ$_{i}$ ) g$_{i}$ − $_{t$_{i}$}$e − $^{Nm$_{i}$}$( Nm$_{i}$ ) g$_{i}$ g$_{i}$ ! × m$_{i}$ α$_{i}$ − $^{1}$e − γ$_{2}$$^{N}$N γ$_{1}$ − 1 [ m$_{i}$ | t$_{i}$, g$_{i}$, N, α , γ$_{1}$, γ$_{2}$ ] ∼ Γ( g$_{i}$ + α$_{i}$, N ) [ g$_{i}$ | T , m , N, α , γ$_{1}$, γ$_{2}$ ] ∼ t$_{i}$ + P oisson ( Nm$_{i}$ (1 − φ$_{i}$ )) [ N | T , g , m , α , γ$_{1}$, γ$_{2}$ ] ∼ Γ( k ∑ i =1 g$_{i}$ + γ$_{1}$, 1 + γ$_{2}$ )