AI 中文总结
研究如何模拟相关二元结果，通过直接在联合伯努利概率质量函数上制定问题，利用线性规划施加多种约束，给出精确PMF公式，还开发了截断矩完成等方案，提供了可行性和模拟框架，明确高斯阈值构造局限。
AI 中文摘要
在风险建模中，模拟具有规定均值和成对皮尔逊相关性的相依伯努利结果是常见任务。常用的高斯阈值工作流程通常被视为高斯copula构造的伯努利类似物。本文表明，阈值化后将潜在高斯相关性设为目标伯努利相关性通常不正确，成对四分体校准仅在校准后的潜在矩阵为半正定才精确。因此直接在联合伯努利概率质量函数上制定问题，通过线性规划施加归一化、非负性、均值约束和成对交叉矩约束。所得的PMF公式要么返回与所需一阶和二阶矩匹配的精确分布，要么证明不可行。还开发了截断矩完成方案和稀疏支持工作集细化方法。这些构造为适度维度和结构化替代方案提供了基于精确PMF的可行性和模拟框架，同时阐明了高斯阈值构造的局限性。
英文摘要
Simulating dependent Bernoulli outcomes with prescribed means and pairwise Pearson correlations is a common task in risk modeling. A familiar approach is the Gaussian-threshold workflow for binary outcomes, often viewed as a Bernoulli analogue of the Gaussian copula construction. We show that setting latent Gaussian correlations equal to target Bernoulli correlations is generally incorrect after thresholding, and that pairwise tetrachoric calibration is exact only when the calibrated latent matrix is positive semidefinite. We therefore formulate the problem directly over the joint Bernoulli probability mass function. Given target means and pairwise correlations, we impose normalization, nonnegativity, mean constraints, and pairwise cross-moment constraints as a linear program over the $2^N$ atomic probabilities. The resulting PMF formulation either returns an exact law matching the requested first and second moments or certifies infeasibility. A convex-hull characterization further shows that every feasible target admits a law supported on at most $1+N+\binom{N}{2}$ states, while every infeasible target admits a separating quadratic certificate. We then develop a truncated-moment completion scheme that fits a reduced cross-moment table and generates samples by sequential conditioning, together with a sparse-support working-set refinement that can reduce memory usage on structured instances, although the worst-case complexity remains exponential. Together, these constructions provide an exact PMF-based framework for feasibility and simulation at moderate dimension and structured alternatives when the full atomic representation is impractical, while clarifying the limits of Gaussian-threshold constructions.
