Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

最近的研究在模拟退火的思想基础上发展了几种蒙特卡洛方法，用于估计归一化常数（分区函数）。本文通过评估多个设计选择对于渐近估计误差的影响，提出了一种两步估计器以高效地近似最优路径。

证明退火对估计归一化常数的益处：重要抽样、噪音对比估计及其它