Parametric families on large binary spaces

In the context of adaptive Monte Carlo algorithms, we cannot directly generate independent samples from the distribution of interest but use a proxy which we need to be close to the target. Generally, such a proxy distribution is a parametric family on the sampling spaces of the target distribution. For continuous sampling problems in high dimensions, we often use the multivariate normal distribution as a proxy for we can easily parametrise it by its moments and quickly sample from it. Our objective is to construct similarly flexible parametric families on binary sampling spaces too large for exhaustive enumeration. The binary sampling problem is more difficult than its continuous counterpart since the choice of a suitable proxy distribution is not obvious.

💡 Research Summary

The paper addresses a fundamental challenge in adaptive Monte Carlo (AMC) methods: when the target distribution cannot be sampled directly, a surrogate “proxy” distribution must be employed, and the quality of this proxy critically determines the efficiency of the entire algorithm. In continuous high‑dimensional settings the multivariate normal distribution serves this role because its parameters (mean vector and covariance matrix) are easy to estimate and sampling is computationally cheap. The authors ask the analogous question for binary spaces of the form ({0,1}^d), where the combinatorial explosion makes exhaustive enumeration impossible and the choice of a suitable parametric family is far from obvious.

Three families are proposed. The simplest is an independent Bernoulli model, parameterised by a vector of marginal probabilities (p_i). While trivially scalable, it cannot capture any dependence among bits. To introduce correlations, the authors adopt an Ising‑type energy function: \