Probabilistic models in physics often require from the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Getting the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. This problem is also present in probabilistic learning models such as the Restricted Boltzmann Machine (RBM), where the situation is even worse as the exact learning rules implies the computation of Z at each iteration. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. So far, the standard application of the AIS algorithm starts from the uniform probability distribution and uses a large number of Monte Carlo steps to obtain reliable estimations of Z following an annealing process. In this work we show that both the quality of the estimation and the cost of the computation can be significantly improved by using a properly selected mean-field starting probability distribution. We perform a systematic analysis of AIS in both small- and large-sized problems, and compare the results to exact values in problems where these are known. As a result of our systematic analysis, we propose two successful strategies that work well in all the problems analyzed. We conclude that these are good starting points to estimate the partition function with AIS with a relatively low computational cost.

翻译：物理学中的概率模型通常需要计算归一化的玻尔兹曼因子，这进而要求计算配分函数Z。然而，随着系统规模增大，精确计算Z变得极其昂贵。这一问题同样存在于受限玻尔兹曼机（RBM）这类概率学习模型中，且情况更为严峻，因为精确学习规则需要在每次迭代中计算Z。解决该问题的可能途径之一是使用退火重要性采样（AIS）算法，该算法提供了一种随机估计系统配分函数的工具。目前，AIS算法的标准应用从均匀概率分布出发，通过大量蒙特卡洛步骤在退火过程中获得Z的可靠估计。本研究表明，通过选择经过适当处理的平均场初始概率分布，可以显著提升估计质量并降低计算成本。我们对AIS在小型和大型问题中进行了系统性分析，并将结果与已知精确值的问题进行对比。基于系统性分析的结果，我们提出了两种在所有分析问题中均表现良好的有效策略。我们得出结论：这些策略是以相对较低的计算成本通过AIS估计配分函数的良好起点。