to squared error loss across all categories, they tend to shrink categories where large counts are observed excessively in order to boost the estimated probabilities of cells with few or no counts. Due to the massive number of categories and the extreme imbalance in observed frequencies (e.g. more than half of the categories have zero counts in our data), the estimates for frequently observed genes tend to be fairly poor. This observation highlights the main weakness of shrinkage estimators, while they perform better on average across categories, they may perform poorly on particular categories which may be of primary interest. Morris et al. (2003) resolve the problem by introducing a hierarchical Dirichlet-Multinomial model which clusters genes into two categories, high and low frequency, partitioning the overall mass in order to mitigate the shrinkage in the high frequency categories. Our contribution is clearly distinct from that of Morris et al. (2003) since it focuses on a discrepancy in the sampling model. Nevertheless, it should be possible to extend our methods to allow for the mixture priors used by Morris et al. (2003). Even without such an extension, our results offer alternative methods and important insights into shrinkage in models with large or massive numbers of categories. For example, we’ve shown that by using the prior α = 1 /k , the posterior mean is equal to the corrected MLE. The additional advantage of a sampling model is that it generates marginal posterior credible sets for the proportion m$_{i}$ . If inference focuses only on genes with large expression levels, we gain one of the main advantages of Monte Carlo sampling without the concern of excessive shrinkage. If one is only interested in point estimates, and excessive shrinkage is a concern, it may also be appropriate to use the analytical mode as an estimator. For example, the posterior mode for model 5 is identical to the MLE for model 3 when α = 1 , while the posterior mean becomes equivalent when α = 1 /k . Given the stated hyper-parameters, posterior means are shrunk for more frequently observed categories and increased for rarely observed categories. The models provide different levels of shrinkage, with the Dirichlet-Poisson-Binomial (DPB) model providing the least, and the missing data model providing the most shrinkage. In 24