We explore a homotopy sampling procedure and its generalization, loosely based on importance sampling, known as annealed importance sampling. The procedure makes use of a known probability distribution to find, via homotopy, the unknown normalization of a target distribution, as well as samples of the target distribution. In the context of stationary distributions that are associated with physical systems the method is an alternative way to estimate an unknown microcanonical ensemble. We make the connection between the homotopy and a dynamics problem explicit. Further, we propose a reformulation of the method that leads to a rejection sampling alternative. We derive the error incurred in computing the target distribution normalization, when sample inversion is not possible. The error in the procedure depends on the errors incurred in sample averaging and the number of stages used in the computational implementation of the process. However, we show that it is possible to exchange the number of homotopy stages and the total number of samples needed at each stage in order to enhance the computational efficiency of the implemented algorithm. Estimates of the error as a function of stages and sample averages are derived. These could guide computational efficiency decisions on how the calculation would be mapped to a given computer architecture. Consideration is given to how the procedure can be adapted to Bayesian estimation problems, both stationary and non-stationary. The connection between homotopy sampling and thermodynamic integration is made. Emphasis is placed on the non-stationary problems, and in particular, on a sequential estimation technique know

中文翻译：

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用同伦法计算概率密度，以及在粒子滤波器中的应用

我们探索了同伦抽样过程及其泛化，松散地基于重要性抽样，称为退火重要性抽样。该过程利用已知概率分布通过同伦找到目标分布的未知归一化，以及目标分布的样本。在与物理系统相关的平稳分布的上下文中，该方法是估计未知微正则系综的另一种方法。我们明确了同伦和动力学问题之间的联系。此外，我们建议重新制定导致拒绝抽样替代方法的方法。当样本反演不可能时，我们推导出计算目标分布归一化时产生的误差。该过程中的误差取决于样本平均中产生的误差以及该过程的计算实现中使用的阶段数。然而，我们表明可以交换同伦阶段的数量和每个阶段所需的样本总数，以提高所实现算法的计算效率。得出了作为阶段和样本平均值函数的误差估计值。这些可以指导关于如何将计算映射到给定计算机体系结构的计算效率决策。考虑了如何使该过程适应贝叶斯估计问题，包括平稳和非平稳。建立了同伦采样和热力学积分之间的联系。重点放在非平稳问题上，特别是，