[Paper Review] Extending Owen's integral table and a new multivariate Bernoulli distribution

Marcelo Hartmann|arXiv (Cornell University)|Apr 16, 2017

Statistical Distribution Estimation and Applications3 references3 citations

TL;DR

This paper extends Owen's integral table by deriving a new multivariate Gaussian integral identity involving products of standard normal CDFs and a normal density, enabling closed-form computation of multivariate Bernoulli distributions. The key contribution is a novel multivariate Bernoulli model where success probabilities are determined by latent Gaussian variables, with marginal likelihoods expressible as multivariate normal CDFs with structured covariance matrices.

ABSTRACT

We extend some equatilites in the Owen's table of normal integrals (A table of normal integrals in Communication in Statistics-Simulation and Computation, 1980). Furthermore a new probabilistic model for a vector of binary random variables is proposed.

Motivation & Objective

• To extend Owen’s integral table by deriving new closed-form solutions for multivariate Gaussian integrals involving standard normal CDFs.
• To develop a new multivariate Bernoulli distribution where success probabilities are linked to latent Gaussian variables via the standard normal CDF.
• To provide analytical solutions for integrals that previously required numerical quadrature in applications such as Gaussian process models and probit regression.
• To establish a connection between multivariate Gaussian CDFs and the marginal likelihoods of discrete latent variable models.

Proposed method

• Derives a general identity for the integral of a product of N standard normal CDFs multiplied by a normal density, showing it equals a multivariate normal CDF with a specific covariance matrix.
• Uses a change of variables transformation to re-express the joint density in terms of auxiliary variables, enabling the identification of a multivariate normal structure.
• Applies the partitioned matrix inversion and determinant lemmas to derive the covariance matrix of the resulting multivariate normal distribution.
• Introduces a new multivariate Bernoulli model where each binary outcome depends on a latent Gaussian variable through the standard normal CDF.
• Transforms the joint distribution of latent variables and binary outcomes using Jacobian adjustment, leading to a marginal distribution expressed as a multivariate normal CDF.
• Establishes that the marginal probability of a binary vector is a multivariate normal CDF with mean −I_y μ and covariance I_N + I_y Σ I_y.

Experimental results

Research questions

• RQ1 Can Owen’s integral table be extended to include multivariate integrals involving products of standard normal CDFs and a normal density?
• RQ2 What is the analytical form of the integral ∫ℝ ∏_{r=1}^N Φ((x−m_r)/v_r) N(x|μ,σ²) dx?
• RQ3 Can a new multivariate Bernoulli distribution be derived from a hierarchical Gaussian-Bernoulli model with latent Gaussian variables?
• RQ4 What is the marginal distribution of a multivariate binary vector when each component is conditionally independent Bernoulli with success probability Φ(f_r), where f_r is Gaussian?
• RQ5 How can the resulting marginal likelihoods be expressed in terms of multivariate normal CDFs with structured covariance matrices?

Key findings

• The integral ∫ℝ ∏_{r=1}^N Φ((x−m_r)/v_r) N(x|μ,σ²) dx equals F_N(μ 1_N | m_N, V_N), where V_N has diagonal elements v_r² + σ² and off-diagonal elements σ².
• The marginal distribution of a multivariate binary vector Y, where Y_r | f_r ~ Bernoulli(Φ(f_r)) and f ~ N(μ, Σ), is π_Y(y) = F_N(0 | -I_y μ, I_N + I_y Σ I_y).
• For N=2, μ = [0,0]^T, σ₁²=σ₂²→0, σ₁₂=1/2, the probabilities are π(1,1)=π(-1,-1)=1/3 and π(1,-1)=π(-1,1)=1/6, consistent with known bivariate normal orthant probabilities.
• The method provides a closed-form solution for integrals that were previously computed via numerical quadrature, improving computational efficiency in Bayesian models.
• The derived multivariate Bernoulli distribution is valid for any N and any mean and covariance structure of the latent Gaussian variables.
• The transformation via z = I_y f and Jacobian adjustment confirms the marginal distribution is a multivariate normal CDF with mean −I_y μ and covariance I_N + I_y Σ I_y.