阶约束非IID有序随机变量的受控精度吉布斯抽样
Controlled Accuracy Gibbs Sampling of Order Constrained Non-IID Ordered Random Variates
摘要：
从$m$独立但不相同分布的随机变量产生的顺序统计量通常通过以递增顺序排列一些$X_{1}, X_{2}, \ldots, X_{m}$来构造，其中$X_{i}$具有分布函数$F_{i}(x)$，表示为$X_{(1)} \leq X_{(2)} \leq \ldots \leq X_{(m)}$。在这种情况下，$X_{(i)}$不一定与$F_{i}(x)$相关联。假设可以模拟来自每个分布的值，则可以通过从$F_{i}$模拟$X_{i}$（对于$i=1,2,\ldots, m$）并按顺序排列它们来生成这种“非iid”顺序统计量。本文考虑模拟有序值$X_{(1)}, X_{(2)}, \ldots, X_{(m)}$使得$X_{(i)}$的边际分布为$F_{i}(x)$的问题。这个问题出现在贝叶斯主成分分析（BPCA）中，其中$X_{i}$是有序的特征值，这些特征值是后验独立的，但不是同分布的。我们提出了一种新的耦合从过去的算法“完美”（可计算的精度顺序）模拟这样的{ \emph{order-constrained non-iid} }顺序统计。我们证明了我们的方法的有效性的几个例子，包括BPCA问题。
英文摘要：
Order statistics arising from $m$ independent but not identically distributed random variables are typically constructed by arranging some $X_{1}, X_{2}, \ldots, X_{m}$, with $X_{i}$ having distribution function $F_{i}(x)$, in increasing order denoted as $X_{(1)} \leq X_{(2)} \leq \ldots \leq X_{(m)}$. 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 $i=1,2,\ldots, m$, and arranging them in order. In this paper, we consider the problem of simulating ordered values $X_{(1)}, X_{(2)}, \ldots, X_{(m)}$ 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. We propose a novel coupling-from-the-past algorithm to "perfectly" (up to computable order of accuracy) simulate such {\emph{order-constrained non-iid}} order statistics. We demonstrate the effectiveness of our approach for several examples, including the BPCA problem.
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