A Practical Guide to Simulating Correlated Binary Outcomes

2026-07-18

AI Summary

This work addresses the challenge of accurately simulating Bernoulli random variables with prescribed means and pairwise Pearson correlations, circumventing the bias inherent in Gaussian copula–based thresholding methods. The authors propose modeling directly on the joint probability mass function, enforcing normalization, non-negativity, and exact first- and second-moment constraints via linear programming to either achieve precise moment matching or certify infeasibility. Key contributions include a proof that any feasible solution is supported on at most $1 + N + \binom{N}{2}$ atomic states, a quadratic separation certificate for detecting infeasible correlation specifications, and algorithmic strategies involving truncated moment completion and sparse support optimization. The approach enables exact simulation and feasibility verification in moderate dimensions, exposes limitations of Gaussian thresholding, and yields memory-efficient approximations for large-scale instances.

Abstract

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.

Problem

• correlated binary outcomes
• Bernoulli simulation
• feasibility
• moment constraints
• tetrachoric correlation

Innovation

• correlated binary outcomes
• joint Bernoulli distribution
• linear programming
• moment constraints
• tetrachoric correlation