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.

翻译：粒子方法包含多种技术，如马尔可夫链蒙特卡洛（MCMC）和序列蒙特卡洛（SMC），通过一组加权粒子来逼近概率目标分布。本文证明，对于任意粒子集合，当目标分布为离散分布时，存在一种唯一的加权机制，能够最小化（基于粒子的）逼近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）相对于该散度度量都是次优的。我们的证明除要求目标分布具有离散性外，无需对目标分布或粒子生成过程施加任何限制。我们证明，最优权重可以根据任何现有粒子方法已计算的值确定；因此，通过最小程度的修改且无需额外计算成本，即可提升任意粒子方法的性能。我们在离散分布的重要应用场景中进行了实证评估，包括贝叶斯变量选择和贝叶斯结构学习。结果表明，我们提出的粒子重加权方法能够一致且显著地改进任意粒子方法对目标分布的逼近效果。