Discrete distributions are learnable from metastable samples

Abstract

Physically motivated stochastic dynamics are widely used to sample from high-dimensional distributions. However, such samplers often get trapped in metastable states, approximately sampling from a distribution that differs significantly from the desired stationary state. We rigorously show that for multivariable discrete distributions, the true stationary model can nevertheless be recovered from these metastable samples. This relies on a fundamental observation: for distributions satisfying a strong metastability condition, their single-variable conditional probabilities are on average extremely close to those of the true stationary distribution. This remains true even when the two distributions are far apart under global metrics such as Kullback-Leibler divergence. Consequently, we can effectively learn the true model using a conditional-likelihood estimator even when the samples are drawn from a restricted state space. Extending these general results to Ising models, we prove rigorous parameter and structure learning guarantees. Finally, we demonstrate this phenomenon numerically on higher-alphabet spin glass models.