which offers improved estimates for proportions with low tag counts while underestimating the expression proportions for tags with large counts. This leads them to propose a mixture Dirichlet prior in order to mitigate the propensity to underestimate highly expressed genes. 

An alternative analysis, which was not based on a multinomial model directly, was de- veloped by Thygesen & Zwinderman (2006), who modeled the marginal distribution of the counts across tag types as though they were independent observations from a Poisson dis- tribution. They applied a hierarchical zero-truncated Poisson model with mean parameter which followed either a gamma or log-normal distribution. A non-parametric adjustment factor was required in order to correctly capture the overabundance of larger counts. Similar analysis (Kuznetsov et al., 2002; Kuznetsov, 2006) modeled SAGE counts using a discrete Pareto-like distribution. These studies found that this model effectively predicted counts greater than zero. One drawback is that the variance of expression cannot be explicitly separated from sample variance. However, in the context of differential expression, Baggerly et al. (2003) suggests that treating genes individually offers less power than a model, such as the multinomial, which incorporates all tags simultaneously. 

One common thread is the explicit assumption that an mRNA’s frequency in the sampled tag pool is equivalent to its frequency in the mRNA population from which the tag pool was derived. However, because the ability to form tags from an mRNA transcript varies from gene to gene, the tag pool that is sampled is actually a biased sample of the mRNA population (Gilchrist et al., 2007). Gilchrist et al. (2007) illustrated how the tag formation bias could be estimated and incorporated through the calculation of a gene specific tag formation probability. The complicating factor is that the sampling bias of one gene is not only a function of its own tag formation probability, but is also inversely proportional to a mean tag formation probability where the contribution of each gene to this mean is weighted by its frequency in the mRNA population, i.e. the very parameters we wish to estimate. 

As a result of the sampling bias, the probability of observing an individual gene depends 

