SIMULATING RATIOS OF NORMALIZING CONSTANTS VIA A SIMPLE IDENTITY: A THEORETICAL EXPLORATION
SIMULATING RATIOS OF NORMALIZING CONSTANTS VIA A SIMPLE IDENTITY: A THEORETICAL EXPLORATION
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发表时间：
1996
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通讯作者：
X. Meng;W. Wong
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作者：
X. Meng;W. Wong
设pi（w），i =1，2，是具有共同支持的两个密度，其中每个密度已知达到归一化常数：pi（w）= qi（w）/ci。通过马尔可夫链蒙特卡罗），我们想使用这些绘制来模拟归一化常数的比率，c1/c2。这样的计算问题经常在似然性和贝叶斯推理中遇到，并且出现在物理学和遗传学等领域。统计学和其他文献中提出的许多方法（例如，计算物理学）处理这个问题的方法都是基于以下简单恒等式的各种特殊情况：c1 c2 = E2（q1（w）α（w））E1（q2（w）α（w））。这里Ei表示相对于pi（i =1，2）的期望值，α是一个任意函数，使得分母不为零。本文的主要目的是提供一个理论研究的有用性，这个身份，重点是（渐近）最佳和实际的选择α。使用一个简单但有启发性的例子，我们证明，与传统的重要性抽样方法（对应于α =1 /q2）相比，通过合理（不一定是最佳）的α选择，我们可以将模拟误差降低几个数量级。我们还介绍了这个身份的几个概括，用于处理更复杂的设置（例如，同时估计几个比率），并提出了几个似乎具有实际和理论价值的开放性问题。此外，我们讨论了相关的理论和实证工作。
Let pi(w) ,i =1 , 2, be two densities with common support where each density is known up to a normalizing constant: pi(w )= qi(w)/ci .W e have draws from each density (e.g., via Markov chain Monte Carlo), and we want to use these draws to simulate the ratio of the normalizing constants, c1/c2. Such a compu- tational problem is often encountered in likelihood and Bayesian inference, and arises in fields such as physics and genetics. Many methods proposed in statistical and other literature (e.g., computational physics) for dealing with this problem are based on various special cases of the following simple identity: c1 c2 = E2(q1(w)α(w)) E1(q2(w)α(w)) . Here Ei denotes the expectation with respect to pi (i =1 , 2), and α is an arbitrary function such that the denominator is non-zero. A main purpose of this paper is to provide a theoretical study of the usefulness of this identity, with focus on (asymptotically) optimal and practical choices of α. Using a simple but informa- tive example, we demonstrate that with sensible (not necessarily optimal) choices of α, we can reduce the simulation error by orders of magnitude when compared to the conventional importance sampling method, which corresponds to α =1 /q2. We also introduce several generalizations of this identity for handling more compli- cated settings (e.g., estimating several ratios simultaneously) and pose several open problems that appear to have practical as well as theoretical value. Furthermore, we discuss related theoretical and empirical work.