The full conditional distributions are derived to be, In order to apply Gibbs Sampler a value for the hyperparameter µ must be selected. In the case where all α$_{i}$ = 1, a logical mean for the Poisson is T$_{tot}$ ∑ i m$_{i}$ (1 − φ$_{i}$ ) / ∑ i m$_{i}$φ$_{i}$ since $^{∑}$m$_{i}$ (1 − φ$_{i}$ ) = 1 − $^{∑}$m$_{i}$ φ$_{i}$ is the probability that an mRNA is not converted into a tag. Equation 4 provides a useful estimate of $^{∑}$m$_{i}$ φ$_{i}$ . Like the DPB algorithm above, the missing data algorithm also admits a simple Lindley- Smith optimization procedure in order to compute the posterior mode. If the prior mean of r , µ = 10 , 100 then the posterior mode of the missing data model is nearly identical to the exact analytical mode. The mode of the Dirichlet conditional is see Gelman et al. (2005). The mode for the Poisson is the mean rounded down to the nearest smaller integer (Wikipedia, 2007). Results for this case are very similar to the previous two. This algorithm is clearly the fastest, completing 5 million samples in 20 minutes. Autocorrelation in the untagged population size r decreases to 0 after 50 simulations but proportions m$_{i}$ seem uncorrelated at all lags. Conclusions for Figures 6 and 7 are largely identical to the above. In fact, as implemented quantitative differences do exist between the methods in the case where all α$_{i}$ = 1 with this method providing the smallest or most heavily shrunk estimates of genes with large counts and the DPB method giving the largest estimates. Because probabilities must sum to one, the opposite effect should exist for categories with 22 ̂ m$_{i}$φ$_{i}$ = t$_{i}$ + α$_{i}$ − 1 $^{∑}$$_{i}$( α$_{i}$ + t$_{i}$ ) + r − k , (7) [ r | T , m , α , µ ] = P oisson ((1 − $^{∑}$m$_{i}$ φ$_{i}$ ) µ ) [ m φ | T , r, φ , α , µ ] = D ( r, α$_{1}$ + t$_{1}$, . . . , α$_{k}$ + t$_{k}$ )