2.4 Comparison to Other Methods 2.4.1 Rejection Sampling 2.4.2 Direct Sampling At first glance, our method looks quite similar to standard rejection sampling. With rejection sampling, one repeatedly samples both a threshold value from a standard uniform distribution ( p ( u ) = 1), and a proposal from g ($_{q}$) until $_{p}$($_{q}$ | y ) / g ($_{q}$) ≥ Ku , where K is a positive constant. This is different from our method, for which the threshold values are sampled from a posterior p ( u | y ) , and K is specifically defined as the ratio of modal densities of the posterior to the proposal. The advantages that rejection sampling has over our approach is that the distribution of u is exact, and that the proposal density does not have to dominate the target density for all values of q . However, for any model with more than a few dimensions, the critical ratio can be extremely small for even small deviations of q away from the mode. Thus, the acceptance probabilities are virtually nil. In contrast, we accept a discrete approximation of p ( u | y ) in exchange for higher acceptance probabilities. Walker et al. (2011) introduced and demonstrated the merits of a non-MCMC approach called Direct Sampling for conducting Bayesian inference. Like our method, Direct Sampling removes the need to concern oneself with issues like chain convergence and autocorrelation, and gener- ates independent samples from a target posterior distribution in parallel . Walker et al. (2011) also prove that the sample acceptance probabilities using Direct Sampling are better than those from standard rejection algorithms. Put simply, for many common Bayesian models, they demonstrate improvement over MCMC in terms of efficiency, resource demands and ease of implementation. However, Direct Sampling suffers from some important shortcomings that limit its broad ap- plicability. One is the failure to separate the specification of the prior from the specifics of the estimation algorithm. Another is an inability to generate accepted draws for even moderately sized problems; the largest number of parameters that Walker et al. consider is 10. Our method allows us to conduct full Bayesian inference on hierarchical models in high dimensions, with or without conjugacy, without MCMC. Although we share some important features with Direct Sampling, our method differs in sev- 10