Simulating normalizing constants: from importance sampling to bridge sampling to path sampling

TLDR

Computing normalizing constants of probability models is a fundamental problem, and Monte Carlo simulation is an effective technique, especially for complex, high‑dimensional models. The paper aims to introduce general statistical audiences to effective methods from theoretical physics and to examine them from a statistical perspective. It establishes theoretical connections and illustrates the methods’ use with statistical problems. The authors demonstrate that acceptance‑ratio and thermodynamic‑integration methods generalize importance sampling via single and continuous bridge densities, respectively, and that their path‑sampling formulation offers greater flexibility and potential efficiency, as illustrated by theoretical and high‑dimensional empirical examples, while also highlighting open problems.

Abstract

Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, especially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to explore these methods from a more statistical perspective, through establishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong. Thermodynamic integration, which is also known in the numerical analysis literature as Ogata's method for high-dimensional integration, corresponds to the use of infinitely many and continuously connected bridges (and thus a "path"). Our path sampling formulation offers more flexibility and thus potential efficiency to thermodynamic integration, and the search of optimal paths turns out to have close connections with the Jeffreys prior density and the Rao and Hellinger distances between two densities. We provide an informative theoretical example as well as two empirical examples (involving 17- to 70-dimensional integrations) to illustrate the potential and implementation of path sampling. We also discuss some open problems.