# Perfect Gibbs Sampling of Order Constrained Non-IID Ordered Random Variates with Application to Bayesian Principal Components Analysis

Order statistics arising from m independent but not identically distributed random variables are typically constructed by arranging some X_1, X_2, …, X_m, with X_i having distribution function F_i(x), in increasing order denoted as X_(1)≤ X_(2)≤…≤ X_(m). In this case, X_(i) is not necessarily associated with F_i(x). Assuming one can simulate values from each distribution, one can generate such “non-iid" order statistics by simulating X_i from F_i, for i=1,2,…, m, and simply putting them in order. In this paper, we consider the problem of simulating ordered values X_(1), X_(2), …, X_(m) such that the marginal distribution of X_(i) is F_i(x). This problem arises in Bayesian principal components analysis (BPCA) where the X_i are ordered eigenvalues that are a posteriori independent but not identically distributed. In this paper, we propose a novel coupling-from-the-past algorithm to “perfectly" (up to computable order of accuracy) simulate such "order-constrained non-iid" order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem.

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