[Paper Review] Characterization of multivariate Bernoulli distributions with given margins

Roberto Fontana, Patrizia Semeraro|arXiv (Cornell University)|Jun 5, 2017

Bayesian Methods and Mixture Models10 references3 citations

TL;DR

This paper proposes a novel method to characterize multivariate Bernoulli distributions with given marginal probabilities by representing the Fréchet class as the convex hull of ray densities. It enables compatibility checking of a target correlation matrix with given margins and provides a linear programming solution to construct a valid joint density, with explicit correlation bounds derived algebraically.

ABSTRACT

We express each Fréchet class of multivariate Bernoulli distributions with given margins as the convex hull of a set of densities, which belong to the same Fréchet class. This characterisation allows us to establish whether a given correlation matrix is compatible with the assigned margins and, if it is, to easily construct one of the corresponding joint densities. % Such %representation is based on a polynomial expression of the distributions of a Fréchet class. We reduce the problem of finding a density belonging to a Fréchet class and with given correlation matrix to the solution of a linear system of equations. Our methodology also provides the bounds that each correlation must satisfy to be compatible with the assigned margins. An algorithm and its use in some examples is shown.

Motivation & Objective

• To address the long-standing challenge of determining whether a given correlation matrix is compatible with specified Bernoulli marginal distributions in multivariate binary data.
• To provide a systematic method for constructing a valid joint density when the correlation matrix and margins are compatible.
• To derive explicit bounds on pairwise correlations that are compatible with the given margins, applicable to any number of variables and correlation structures.
• To develop a computational framework for simulating multivariate binary data with prespecified margins and moments, using polynomial and matrix representations.

Proposed method

• Represent all multivariate Bernoulli distributions in a Fréchet class as the convex hull of ray densities, which are extreme points of the class.
• Use a polynomial and matrix representation based on the Kronecker product of difference matrices to express the joint density in terms of cumulative distribution functions.
• Formulate the problem of finding a joint density with given moments as a system of linear equations involving the ray matrix and correlation constraints.
• Construct the ray matrix $ R_p $ using the 4ti2 software to enumerate all extreme distributions in the Fréchet class.
• Solve the resulting linear system using standard linear programming tools (e.g., SAS/IML and Proc Lpsolve) to find a valid density satisfying marginal and correlation constraints.
• Derive analytical bounds on pairwise correlations using the structure of the ray matrix and the linear system, ensuring compatibility with the given margins.

Experimental results

Research questions

• RQ1 Can a given correlation matrix be compatible with a set of specified Bernoulli marginal distributions in multivariate binary data?
• RQ2 What is the set of all possible joint distributions that are consistent with fixed Bernoulli margins and a given correlation structure?
• RQ3 How can one efficiently construct a valid joint density for multivariate binary data when the correlation matrix and margins are compatible?
• RQ4 What are the theoretical bounds on pairwise correlations that are compatible with a given set of marginal probabilities?
• RQ5 Can the simulation of multivariate binary data with prespecified margins and moments be reduced to solving a linear system of equations?

Key findings

• The Fréchet class of multivariate Bernoulli distributions with given margins is the convex hull of a finite set of ray densities, enabling exact characterization.
• For a trivariate case with margins $ p = (1/4, 1/4, 1/3) $, the correlation bounds are $ -0.236 \leq \rho_{12} \leq 0.707 $, $ -0.408 \leq \rho_{13} \leq 0.816 $, and $ -0.289 \leq \rho_{23} \leq 0.577 $.
• When $ \rho_{12} = 0.3 $, $ \rho_{13} = 0.25 $, and $ \rho_{23} = -0.2 $, a valid joint density was constructed with $ \boldsymbol{f}_p^T = (0.0146, 0, 0.1197, \dots, 0.4893) $.
• For $ m = 5 $ with all margins $ p_i = 1/2 $, the method successfully generated a valid density with 2,712 ray densities and a specified correlation structure.
• For $ m = 6 $ with all $ p_i = 1/2 $, the method handled 707,264 ray densities, demonstrating scalability under computational constraints.
• The approach reduces the simulation of multivariate binary data with given margins and moments to solving a linear system, enabling efficient computation via standard linear programming tools.