[Paper Review] A Generic Multivariate Distribution for Counting Data

Marcos A. Capistrán, J. Andrés Christen|arXiv (Cornell University)|Mar 24, 2011

Bayesian Methods and Mixture Models8 references3 citations

TL;DR

This paper proposes a generic multivariate discrete distribution for counting data defined solely by its mean vector and variance-covariance matrix, using a blend of Poisson, Binomial, and Negative Binomial univariate components combined with a normal copula. The method enables computationally efficient Bayesian inference in complex stochastic models—such as epidemic SIR models—where exact likelihoods are intractable, with the approximation closely matching moments and correlation structure even for low counts.

ABSTRACT

Motivated by the need, in some Bayesian likelihood free inference problems, of imputing a multivariate counting distribution based on its vector of means and variance-covariance matrix, we define a generic multivariate discrete distribution. Based on blending the Binomial, Poisson and Negative-Binomial distributions, and using a normal multivariate copula, the required distribution is defined. This distribution tends to the Multivariate Normal for large counts and has an approximate pmf version that is quite simple to evaluate.

Motivation & Objective

• To develop a default multivariate discrete distribution for counting data when only the mean vector and covariance matrix are known, analogous to the multivariate normal for continuous data.
• To address the challenge of Bayesian likelihood-free inference in stochastic epidemic models where exact likelihoods are computationally infeasible.
• To create a distribution that remains valid and accurate even for low counts, where normal approximations fail.
• To provide a computationally efficient alternative to simulation-based methods like ABC in parameter estimation for epidemic models.

Proposed method

• The univariate component is a mixture of Poisson, Binomial, and Negative Binomial distributions, parameterized by mean μ and variance v, denoted Gd(μ, v).
• The multivariate distribution is constructed using a multivariate normal copula to induce dependence between variables, with margins transformed via inverse cumulative distribution functions.
• The exact multivariate pmf is defined via integration over hypercubes using the multivariate normal density with specified correlation matrix ρ.
• An approximate pmf is derived using a normalization factor K and the ratio of the multivariate normal density to the product of standard normal densities, enabling fast evaluation.
• The transformation s_i = Φ⁻¹(F_μ_i,v_i(x_i)) maps discrete marginals to standard normal variates, preserving correlation structure under the copula.
• The method ensures that the resulting distribution matches the input moments (means, variances, and correlations) and converges to multivariate normality for large counts.

Experimental results

Research questions

• RQ1 Can a generic multivariate discrete distribution be constructed from only the mean vector and covariance matrix, without requiring full model specification?
• RQ2 How can such a distribution be efficiently computed and used as a likelihood in Bayesian inference for stochastic epidemic models?
• RQ3 To what extent does the approximation preserve the true correlation structure and moment matching when counts are low?
• RQ4 How does the performance of the approximate pmf compare to the exact version in terms of computational cost and accuracy?

Key findings

• The proposed distribution, Gd_n, provides a valid multivariate discrete distribution defined exclusively by its mean vector and covariance matrix, analogous to the multivariate normal.
• The approximate pmf gd_n is computationally efficient and closely matches the exact pmf, with contour plots in Figure 1 showing near-identical shapes across all tested cases.
• The normalization constant K in the approximation is consistently close to 1 (e.g., 0.99 or 0.97), suggesting it can be treated as approximately constant in practice.
• The resulting correlation ρ′ from the exact distribution is very close to the target ρ (e.g., 0.5144 vs. 0.5 in case a), confirming accurate dependence structure.
• The approximate distribution matches the first and second moments of the true distribution, with simulated moments μ_i*, v_i*, and ρ* closely aligning with input parameters.
• For large counts, the distribution asymptotically approaches a multivariate normal distribution, as demonstrated in cases (a) and (b) of Figure 1.