On parametric families for sampling binary data with specified mean and correlation

Abstract

We discuss a class of binary parametric families with conditional probabilities taking the form of generalized linear models and show that this approach allows to model high-dimensional random binary vectors with arbitrary mean and correlation. We derive the special case of logistic conditionals as an approximation to the Ising-type exponential distribution and provide empirical evidence that this parametric family indeed outperforms competing approaches in terms of feasible correlations.

Particle Algorithms for Optimization on Binary Spaces

“…are equal the cumulative distribution function of the multivariate normal with respect to the entries indexed by I (see Schäfer (2012) for a more detailed discussion). In particular, the first and second moments are…”

“…Indeed, stronger diagonal dominance in F corresponds to exponential quadratic distributions π(γ) := exp(γ Fγ) γ∈B d exp(γ Fγ) having lower dependencies between the components of γ. We can analytically derive a parameter A ∈ R d×d for a logistic conditionals family q A that approximates π(γ) where the quality of the approximation increases as the diagonal of F becomes more dominant Schäfer (2012). We can accelerate the sequential Monte Carlo algorithm by initializing the system from q A instead of U B d .…”

“…The locally fitted correlation matrices Σ might not be positive definite for d ≥ 3 because the Gaussian copula is not flexible enough to model the full range of crossmoments binary distributions might have (see Schäfer (2012) for an extended discussion). We obtain a feasible parameter replacing Σ by…”