In the classical biased sampling problem, we have k densities π1(·), …, πk (·), each known up to a normalizing constant, i.e. for l = 1, …, k, πl (·) = νl (·)/ml , where νl (·) is a known function and ml is an unknown constant. For each l, we have an iid sample from πl ,·and the problem is to estimate the ratios ml/ms for all l and all s. This problem arises frequently in several situations in both frequentist and Bayesian inference. An estimate of the ratios was developed and studied by Vardi and his co-workers over two decades ago, and there has been much subsequent work on this problem from many different perspectives. In spite of this, there are no rigorous results in the literature on how to estimate the standard error of the estimate. We present a class of estimates of the ratios of normalizing constants that are appropriate for the case where the samples from the πl 's are not necessarily iid sequences, but are Markov chains. We also develop an approach based on regenerative simulation for obtaining standard errors for the estimates of ratios of normalizing constants. These standard error estimates are valid for both the iid case and the Markov chain case.

中文翻译：

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通过再生对来自多个马尔可夫链的归一化常数比率进行估计和标准误差。


在经典的有偏抽样问题中，我们有 k 个密度 π1(·), …, πk (·)，每个密度已知一个归一化常数，即 l = 1, …, k, πl (·) = νl (·) /ml ，其中 νl (·) 是已知函数，ml 是未知常数。对于每个 l，我们有一个来自 πl 的 iid 样本，·问题是估计所有 l 和所有 s 的 ml/ms 比率。这个问题在频率论和贝叶斯推理中的几种情况下经常出现。瓦尔迪和他的同事在二十多年前就对这些比率进行了估计，并进行了研究，随后从许多不同的角度对这个问题进行了大量的研究。尽管如此，文献中还没有关于如何估计估计值的标准误差的严格结果。我们提出了一类归一化常数比率的估计，这些估计适用于 πl 中的样本不一定是独立同分布序列，而是马尔可夫链的情况。我们还开发了一种基于再生模拟的方法，用于获取标准化常数比率估计的标准误差。这些标准误差估计对于独立同分布情况和马尔可夫链情况均有效。