We consider the problem of sampling a high dimensional multimodal target probability measure. We assume that a good proposal kernel to move only a subset of the degrees of freedoms (also known as collective variables) is known a priori. This proposal kernel can for example be built using normalizing flows. We show how to extend the move from the collective variable space to the full space and how to implement an accept-reject step in order to get a reversible chain with respect to a target probability measure. The accept-reject step does not require to know the marginal of the original measure in the collective variable (namely to know the free energy). The obtained algorithm admits several variants, some of them being very close to methods which have been proposed previously in the literature. We show how the obtained acceptance ratio can be expressed in terms of the work which appears in the Jarzynski-Crooks equality, at least for some variants. Numerical illustrations demonstrate the efficiency of the approach on various simple test cases, and allow us to compare the variants of the algorithm.

翻译：本文研究高维多峰目标概率测度的采样问题。我们假设已预先获知一个仅对部分自由度（即集体变量）进行转移的优质提议核，例如该核可通过归一化流构建。我们展示了如何将集体变量空间的转移扩展至完整空间，并如何实施接受-拒绝步骤以获得关于目标概率测度的可逆马尔可夫链。该接受-拒绝步骤无需已知原测度在集体变量上的边缘分布（即无需已知自由能）。所得算法可衍生出多种变体，其中部分与文献中已提出的方法高度相似。我们证明了所得接受率至少在某些变体中可表示为Jarzynski-Crooks等式中所出现的功函数。数值实验通过多个简单测试案例验证了该方法的有效性，并对算法变体进行了比较分析。