Markov Logic in Infinite Domains

Markov logic combines logic and probability by attaching weights to first-order formulas, and viewing them as templates for features of Markov networks. Unfortunately, in its original formulation it does not have the full power of first-order logic, because it applies only to finite domains. Recently, we have extended Markov logic to infinite domains, by casting it in the framework of Gibbs measures. In this talk I will summarize our main results to date, including sufficient conditions for the existence and uniqueness of a Gibbs measure consistent with an infinite MLN, and properties of the set of consistent measures in the non-unique case. (Many important phenomena, like phase transitions, are modeled by non-unique MLNs.) Under the conditions for existence, we have extended to infinite domains the result in Richardson and Domingos (2006) that first-order logic is the limiting case of Markov logic when all weights tend to infinity. I will also discuss some fundamental limitations of Herbrand interpretations (and representations based on them) for probabilistic modeling of infinite domains, and how to get around them. Finally, I will discuss some of the surprising insights for learning and inference in large finite domains that result from considering the infinite limit.