justifies this assumption. 

As mentioned above, in addition to lags of 100 between samples, a very large burn-in period was used in the analyses of Sections 4.1 − 4.3. In order to ensure that autocorrelation did not adversely effect parameter estimates, some basic convergence analysis was performed. Convergence was evaluated on 2 sampled quantities, the total mRNA population size N and the gene at the third locus in the data set: YAL003W. Both quantities have direct inferential value and autocorrelation could adversely affect estimates. YAL003W was selected randomly among the set of genes with medium to large tag counts. Out of the 12799 tags observed, 32 could be attributed to gene YAL003W. Direct autocorrelations of both sequences were tested at various lags. Table 1 examines correlations among samples for different lags and suggests that for inference on proportions m$_{i}$ , autocorrelations are very low between samples with lags as small as 10. Conversely, correlations, for sample size N are extremely high leading to much less reliable inference. 

In practice, if only the vector m is of interest, far fewer samples are needed which would significantly speed up the algorithm. Physical timings for runs of these and other algorithms are given in Table 2. These show that as executed, this algorithm required about 11-12 hours to run. If only every 10th sample were collected, 50,000 total samples, the algorithm would have required approximately 1 hour. 

In order to understand the behavior of the proportions, we compared various estimates of the m$_{i}$ . These include posterior means from the Gibbs sampler, standard MLE’s, corrected MLE’s based on Formula 4, and analytically derived posterior modes from Gilchrist et al. (2007). In addition, 95% posterior credible regions are also computed. Figure 2 plots the above quantities versus rank for the 20 genes with largest tag counts. Rank 1 corresponds to the most frequent gene and the prior used is α = 1 . As one would expect, the corrected MLE and the posterior mode coincide perfectly in this case. Both posterior means and posterior modes based on the biased sampling correction deviate strongly from the standard 

