MLE for many of the most frequently tagged genes. Posterior modes and means follow nearly identical trends with the modal estimates being uniformly larger. As one can see, modal estimates typically lie slightly above the 97 . 5 th percentile of the simulated posterior distribution in this case. Figure 3 summarizes results when each α$_{i}$ = 1 /k . This parametrization corresponds to a U-shaped prior distribution. Here one can see the i th corrected MLE and posterior mean now coincide almost perfectly while the posterior mode lies at or above the 97 . 5 th percentile of posterior samples. Again, all estimators deviate significantly from the unadjusted MLE’s. Due to the similarity of the corrected MLE and posterior mean, it is tempting to interpret the posterior intervals as frequentist confidence intervals. To date, no work has been done to verify the frequentist coverage properties of these intervals. For inferential purposes, the posterior mean vector and both joint and marginal intervals can be derived from the Monte Carlo samples. However, as discussed below in Section 5, it may be useful to compute the joint posterior mode for use as an estimator. For example, the posterior mode reduces to the MLE when the parameter α$_{i}$ = 1 for all i . A second advantage of this model is that the joint mean can be computed through the Lindley-Smith maximization method (Lindley, D. V. & Smith, A. F. M., 1972; Chen et al., 2000). Lindley and Smith proved that an algorithm that iteratively maximized each of the full conditional distributions will eventually converge to the global maximum of the posterior. Because each of the conditionals in this model has a closed form expression for its maximum, this optimization approach is both simple and extremely fast. Similar to Gibbs sampling, this procedure starts from an arbitrary point. Maximization proceeds by iterating across the full set of conditional distributions, maximizing each and substituting the maximum into the next conditional. The hierarchical model considered here is not identical to eq. (3), but depends upon hyperparameters ( γ$_{1}$, γ$_{2}$ ) which effect the mode of the posterior distribution. One possibility for choosing these hyper-parameters is to pick them to minimize the distance 17