离散分布的最优基于粒子的近似（OPAD）
Optimal Particle-based Approximation of Discrete Distributions (OPAD)
AI总结：
针对离散分布的基于粒子近似问题，本文证明存在唯一最小化KL散度的最优加权机制，该机制可利用现有粒子方法已计算值确定，通过重加权能以最小成本持续且显著提升近似性能。
摘要：
基于粒子的方法包括多种技术，如马尔可夫链蒙特卡罗（MCMC）和序贯蒙特卡罗（SMC），用于通过一组加权粒子近似概率目标分布。本文证明，当目标分布为离散分布时，对于任意粒子集，存在唯一的加权机制可最小化（基于粒子的）近似分布与目标分布之间的Kullback-Leibler（KL）散度——任何其他加权机制（例如基于粒子在马尔可夫链中重复次数的MCMC加权）在该散度度量下均为次优。我们的证明除了要求目标分布是离散的外，对目标分布或粒子生成过程没有任何其他限制。我们表明，最优权重可基于现有任何基于粒子的方法已计算的值确定；因此，只需最少修改且无额外计算成本，即可改进任何基于粒子的方法的性能。我们在离散分布的重要应用场景（包括贝叶斯变量选择和贝叶斯结构学习）上进行了实证评估。结果表明，我们提出的粒子重加权方法持续且常显著地改进了任何基于粒子的目标分布近似。
英文摘要：
Particle-based methods include a variety of techniques, such as Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC), for approximating a probabilistic target distribution with a set of weighted particles. In this paper, we prove that for any set of particles, there is a unique weighting mechanism that minimizes the Kullback-Leibler (KL) divergence of the (particle-based) approximation from the target distribution, when that distribution is discrete -- any other weighting mechanism (e.g. MCMC weighting that is based on particles' repetitions in the Markov chain) is sub-optimal with respect to this divergence measure. Our proof does not require any restrictions either on the target distribution, or the process by which the particles are generated, other than the discreteness of the target. We show that the optimal weights can be determined based on values that any existing particle-based method already computes; As such, with minimal modifications and no extra computational costs, the performance of any particle-based method can be improved. Our empirical evaluations are carried out on important applications of discrete distributions including Bayesian Variable Selection and Bayesian Structure Learning. The results illustrate that our proposed reweighting of the particles improves any particle-based approximation to the target distribution consistently and often substantially.
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