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Reminding readers that φ = ( φ$_{1}$, φ$_{2}$, . . . , φ$_{k}$ ) be the vector of known gene dependent tagging probabilities, the full conditional distributions here are, 

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where m ∗ (1 − φ ) is an element wise product with 1 representing a vector of identical dimension to m . Hence, the i th element of the vector is given by m$_{i}$ (1 − φ$_{i}$ ). 

We refer to this hierarchical model as the Dirichlet-Multinomial-Binomial (DMB) model. Inference in this case is much faster because the entire m vector can be updated at once instead of requiring separate updates for each gene level i . This formulation is also more natural in the sense that it restricts the pre-tagging counts to sum to the population total N . One drawback is that the observed data provides no information for posterior inference about the population size N . A minor drawback of this approach is that it is more difficult to perform a Lindley - Smith maximization to derive the mode. Maximizing the conditional distribution of g , the vector of mRNA counts, would require maximizing a multinomial distribution over the discrete counts. 

Experimental testing of this algorithm provided results similar to the first procedure. Five-million variates were drawn with every hundredth stored. The DMB algorithm was dramatically faster than the DPB approach finishing simulation in just over 30 minutes. Table 1 shows little if any correlation at any lag. For sample size N , this is to be expected since the conditional distributions depend only on the data and hyperparameters which are fixed. We conclude that the algorithm could be accelerated without loss of accuracy by reducing the number of simulations. Inference was based on the final 1000 stored samples. Figure 4 below displays inferential results from this model for the 20 most frequently tagged 

