用于采样平衡且紧凑的选区重划方案的序贯蒙特卡洛方法
Sequential Monte Carlo for Sampling Balanced and Compact Redistricting Plans
AI总结：
针对现有选区重划方案采样方法的不足，提出一种可纳入多类现实约束、并行采样的序贯蒙特卡洛算法，经小图验证和宾州案例测试，其收敛速度和样本效率均优于同类马尔可夫链蒙特卡洛算法。
摘要：
在约束下对图划分进行随机采样已成为评估立法选区重划方案的常用工具。分析人员通过将拟议的选区重划方案与采样得到的备选方案集合进行对比，来识别党派不公正划分选区的行为。要实现成功应用，采样方法必须能够扩展到拥有中等或大量选区的地图，纳入符合现实的法律约束，并且能从选定的目标分布中准确高效地采样。遗憾的是，大多数现有方法至少在其中一个方面存在不足。我们提出了一种新的序贯蒙特卡洛（Sequential Monte Carlo, SMC）算法，它能生成收敛于真实目标分布的选区重划方案样本。由于SMC算法可以并行生成大量方案，它比现有的按顺序生成方案的马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）算法能更高效地探索相关的选区重划方案空间。我们的算法可以同时纳入现实选区重划问题中常见的多种约束，包括人口均等、紧凑性以及行政边界保留。我们通过一张可枚举所有选区重划方案的小型地图，验证了所提算法的准确性。随后我们将该SMC算法应用于宾夕法尼亚州近期一起备受关注的选区重划案件中，评估相关方提交的若干份地图的党派影响。我们发现，与同类MCMC算法相比，所提算法收敛速度更快，所需样本量更少。目前已有开源软件可用于实现该方法。
英文摘要：
Random sampling of graph partitions under constraints has become a popular tool for evaluating legislative redistricting plans. Analysts detect partisan gerrymandering by comparing a proposed redistricting plan with an ensemble of sampled alternative plans. For successful application, sampling methods must scale to maps with a moderate or large number of districts, incorporate realistic legal constraints, and accurately and efficiently sample from a selected target distribution. Unfortunately, most existing methods struggle in at least one of these areas. We present a new Sequential Monte Carlo (SMC) algorithm that generates a sample of redistricting plans converging to a realistic target distribution. Because it draws many plans in parallel, the SMC algorithm can efficiently explore the relevant space of redistricting plans better than the existing Markov chain Monte Carlo (MCMC) algorithms that generate plans sequentially. Our algorithm can simultaneously incorporate several constraints commonly imposed in real-world redistricting problems, including equal population, compactness, and preservation of administrative boundaries. We validate the accuracy of the proposed algorithm by using a small map where all redistricting plans can be enumerated. We then apply the SMC algorithm to evaluate the partisan implications of several maps submitted by relevant parties in a recent high-profile redistricting case in the state of Pennsylvania. We find that the proposed algorithm converges faster and with fewer samples than a comparable MCMC algorithm. Open-source software is available for implementing the proposed methodology.
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