We develop a new method to sample from posterior distributions in Bayesian hierarchical models, without using Markov chain Monte Carlo. This method, which is a variant of rejection sampling ideas, is generally applicable to high-dimensional models involving large data sets. Samples are independent, so they can be collected in parallel, and we do not need to be concerned with issues like chain convergence and autocorrelation. The method is scalable under the assumption that heterogeneous units are conditionally independent, and it can also be used to compute marginal likelihoods.
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