退火路径的变分表示：单调嵌入下的Bregman信息
Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding
AI总结：
本文通过单调嵌入下的Bregman散度框架，证明拟算术均值在变换后最小化期望散度，从而揭示退火路径中间密度的变分表示及其与常见散度泛函的关联。
摘要：
马尔可夫链蒙特卡洛方法在从复杂分布中采样以及估计归一化常数时，通常沿着一条退火路径模拟一系列中间分布的样本，该路径在易于处理的初始分布与感兴趣的目标密度之间架起桥梁。已有工作利用拟算术均值构造退火路径，并将所得的中间密度解释为对端点期望散度的最小化。为分析这些退火路径的变分表示，我们扩展了已知结果，即参数上的算术均值最小化到单一代表点的期望Bregman散度。特别地，当Bregman散度的输入在单调嵌入函数下进行变换时，我们获得了拟算术均值的类似结果。我们的分析利用rho-tau表示性Bregman散度框架，凸显了拟算术均值、参数族与散度泛函之间的相互作用，并将常见的散度泛函与沿退火路径的中间密度联系起来。
英文摘要：
Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior works have constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. To analyze these variational representations of annealing paths, we extend known results showing that the arithmetic mean over arguments minimizes the expected Bregman divergence to a single representative point. In particular, we obtain an analogous result for quasi-arithmetic means, when the inputs to the Bregman divergence are transformed under a monotonic embedding function. Our analysis highlights the interplay between quasi-arithmetic means, parametric families, and divergence functionals using the rho-tau representational Bregman divergence framework, and associates common divergence functionals with intermediate densities along an annealing path.
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