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genes ranked from largest to smallest. Behavior and trends are almost identical to the DPB model. For the case where all α$_{i}$ = 1, the posterior mean was systematically smaller than the posterior-mode/MLE for cases with large tag counts. The posterior-mode was distinctly different than the unadjusted tag proportions and both sets were above the upper credible bounds for the posterior mean. In the case where all α$_{i}$ = 1 /k , the posterior-mean again matches the corrected MLE almost exactly. The posterior mode tracks the mean closely but is always larger and the unadjusted proportions are usually outside the credible region. 

## 4.3 A Missing Data Approach. 

Instead of renormalizing the probabilities θ$_{i}$ = ( m$_{i}$φ$_{i}$ ) / ( $^{∑}$m$_{i}$ φ$_{i}$ ) and doing inference based on the posterior of eq. (5), a second more straightforward approach uses missing data. 

Considering again a data generation process as described in section 3.2, we augment the observed data vector T with an extra category which represents any cDNA that is not converted to a SAGE tag. This count is not observed, but if a distribution such as a Poisson is proposed, we may consider a Bayesian approach. The resulting posterior distribution for the data is, 

<img src='content_image/321082.jpg'>

where we have assumed a Dirichlet prior on the unknown initial proportions m ∼ D$_{k}$( α$_{1}$, . . . , α$_{k}$ ) . The resulting posterior distribution is a Dirichlet distribution with k + 1 categories, com- pared with the k categories in the initial distribution m . In particular the distribution is a D$_{k}$$_{+1}$( t$_{1}$ + α$_{1}$, . . . , t$_{k}$ + α$_{k}$, r + 1). 

By integrating over the missing data term or otherwise eliminating it, it is possible to sample from the joint distribution whose success probability is m$_{i}$φ$_{i}$ . According to the marginalization property of the Dirichlet (Bilodeau & Brenner, 1999) the distribution of any subset of variables ( θ$_{i}$$_{1}$, θ$_{i}$$_{2}$, . . . , θ$_{i}$$_{j}$ ) from a D$_{k}$( α$_{1}$, α$_{2}$, . . . , α$_{k}$ ) with $^{∑}$α$_{i}$ = p is distributed 

