---
title: Multivariate Bernoulli Distribution
url: https://www.emergentmind.com/topics/multivariate-bernoulli-distribution
type: topic
---

# Multivariate Bernoulli Distribution

A multivariate Bernoulli distribution is a probability law for a vector $X = (X_1,\dots,X_m)$ of binary random variables, $X_i \in \{0,1\}$, defined on the discrete hypercube $\{0,1\}^m$, with prescribed marginal distributions (each $X_i \sim \text{Bernoulli}(p_i)$) and, possibly, specified dependence structure such as pairwise correlation matrix $(\rho_{ij})$. The Fréchet class of such distributions comprises all joint laws with fixed marginals $p = (p_1,\dots,p_m)$. Characterizing and constructing multivariate Bernoulli laws with given margins and correlations is central to discrete multivariate modeling, combinatorics, and dependent risk simulation [1706.01357].

## 1. Fréchet Class: Definition and Structure

For $m \geq 1$ and margin vector $p = (p_1, \dots, p_m)$ with $0 < p_i < 1$, the Fréchet class is
\[ 
F(p_1,\dots,p_m) = \{\, F \in F_m: X_i \sim \text{Bernoulli}(p_i),\, i=1,\dots,m \,\}
\]
where $F_m$ denotes all distribution functions on $\{0,1\}^m$. This class is equivalently described in terms of joint densities $f: \{0,1\}^m \to [0,1]$ with the constraints:
\[
\sum_{x \in \{0,1\}^m} f(x) = 1, \qquad \sum_{x: x_i=1} f(x) = p_i, \;\; i=1,\dots,m
\]
Thus, $F(p)$ is a convex polytope cut out by the $m$ margin constraints and the simplex condition. This setup encompasses all feasible dependence structures (including both extreme positive/negative association and independence) compatible with the specified margins [1706.01357].

## 2. Convex Geometry and Extremal Representation

Theorem 3.2 in [1706.01357] formally establishes that the set $\mathcal{D}(p)$ of all densities $f$ with margins $p$ is a convex polytope in $\mathbb{R}^{2^m}$, whose extreme points (vertices) can be determined algebraically. Explicitly, every $f \in \mathcal{D}(p)$ is a convex combination
\[
f(x) = \sum_{i=1}^{n_F} \lambda_i R^{(i)}(x)
\]
where $\{R^{(i)}\}_{i=1}^{n_F}$ are the ray densities (vertices or extremal distributions) of the polytope, $\lambda_i \geq 0$, and $\sum_{i} \lambda_i = 1$. These ray densities are computed as nonnegative solutions to the homogeneous margin-constraint system $H r = 0$ (where $H$ is the $m \times 2^m$ margin-constraint matrix), and generally possess support of minimal size dictated by the polytope's combinatorics [1706.01357].

## 3. Polynomial and Copula-Type Expansions

Every $f \in \mathcal{D}(p)$ admits a unique polynomial (Farlie-Gumbel-Morgenstern-type) expansion indexed by subsets $A \subseteq \{1, \dots, m\}$:
\[
f(x) = \left( D^{\otimes m} F \right)(x) = \sum_{A \subseteq \{1,\dots,m\}} \theta_A \prod_{i \in A} (-1)^{1-x_i} p_i^{x_i} q_i^{1-x_i}
\]
where $q_i = 1 - p_i$, $D$ is the finite difference operator, and the parameter vector $(\theta_A)$ encodes dependence at all interaction orders up to $m$ [1706.01357]. The cumulative function $F$ is expressed as a multilinear polynomial in $(1 - u_i)$, generalizing copula representations in the discrete setting:
\[
F(u) = 1 + \sum_{i} \theta_i (1-u_i) + \sum_{i<j} \theta_{ij}(1-u_i)(1-u_j) + \ldots + \theta_{12\cdots m} \prod_{i=1}^m (1-u_i)
\]
This structure mirrors the FGM copula, but, in finite support and with matching margins, the parameters must satisfy linear constraints induced by the margin equations.

## 4. Compatibility and Realizability of Correlation Matrices

For a given pairwise correlation matrix $(\rho_{ij})$, the compatibility problem is: does there exist $f \in \mathcal{D}(p)$ with pairwise second moments matching
\[
\mathbb{E}[X_i X_j] = p_i p_j + \rho_{ij} \sqrt{p_i q_i p_j q_j}
\]
for all $1 \leq i < j \leq m$? Proposition 3.1 in [1706.01357] shows that this is the case if and only if the target vector of second moments $\mu_2$ lies in the convex hull of the second-moment columns of the ray matrix $A_{2,p}$. Explicitly, the feasibility system is
\[
A_{2,p} \lambda = \mu_2, \qquad \sum_i \lambda_i = 1, \quad \lambda_i \geq 0
\]
where $A_{2,p}\in\mathbb{R}^{m(m-1)/2 \times n_F}$ collects the pairwise moments of the rays. If a nonnegative $\lambda$ exists, the desired law can be constructed as $f = \sum_i \lambda_i R^{(i)}$. If not, $\mu_2$ is infeasible, and one may project onto the feasible region to find the nearest compatible correlation structure [1706.01357].

## 5. Bounding Achievable Correlations

Proposition 3.2 gives explicit bounds for each pairwise correlation:
\[
L_{ij} \leq \mathbb{E}[X_i X_j] \leq U_{ij}
\]
where $L_{ij}$ and $U_{ij}$ are the minimal and maximal pairwise moments among all rays. In terms of correlations,
\[
\rho_{ij} \in \left[ \frac{L_{ij} - p_i p_j}{\sqrt{p_i q_i p_j q_j}},\, \frac{U_{ij} - p_i p_j}{\sqrt{p_i q_i p_j q_j}} \right]
\]
The bivariate case reduces to the classical Fréchet–Hoeffding bounds:
\[
\max\{0, p_1 + p_2 - 1\} \leq \mathbb{E}[X_1 X_2] \leq \min\{p_1, p_2\}
\]
This establishes the sharp compatibility region for each pair of margins and attainable correlation [1706.01357].

## 6. Algorithmic Construction and Numerical Illustration

The procedure for constructing a multivariate Bernoulli law with prescribed margins and correlations involves:
1. Building the constraint matrix $H$ for the margin system $H f = 0$;
2. Computing the extremal rays $R^{(i)}$ as generators of the kernel cone (e.g., via 4ti2);
3. Assembling the ray matrix $R_p = [R^{(1)},\dots,R^{(n_F)}]$ and the second-moment matrix $A_{2,p}$;
4. Solving the linear program $A_{2,p} \lambda = \mu_2, \sum \lambda = 1, \lambda \geq 0$ for feasibility;
5. Forming the joint probability vector $f = R_p \lambda$ if a solution exists.

In the case $m=3$, $p_i = 0.5$ for $i = 1,2,3$, and desired $(\rho_{12},\rho_{13},\rho_{23}) = (0.2, -0.3, 0.4)$, the algorithm yields the explicit weight vector $\lambda$ and the corresponding joint pmf on $\{0,1\}^3$ [1706.01357].

## 7. Theoretical and Practical Implications

This convex-geometric characterization unifies the study of discrete dependence, simulation, and parameter compatibility in multivariate Bernoulli laws. Through explicit polytope structure, any marginal/correlation assignment can be checked for feasibility; if feasible, extremal and mixed laws can be synthesized efficiently. The theoretical framework extends naturally to related combinatorial problems, correlation polytopes, and computational methods for high-dimensional discrete data [1706.01357]. The approach is foundational for simulation in risk aggregation, network modeling, and dependence-extremal analysis in statistics and applied probability.

Source: https://www.emergentmind.com/topics/multivariate-bernoulli-distribution