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.

翻译：从复杂分布中采样及估计归一化常数的马尔可夫链蒙特卡洛方法，通常沿退火路径对一系列中间分布进行采样，该路径在易于处理的初始分布与目标密度之间建立桥梁。已有研究通过拟算术均值构造退火路径，并将所得中间密度解释为最小化与端点的期望散度。为分析退火路径的这些变分表示，我们扩展了关于算术均值最小化到单一代表点的期望布雷格曼散度的已知结论。特别地，当布雷格曼散度的输入经单调嵌入函数变换后，我们获得了拟算术均值的类似结论。我们的分析利用rhop-tau表示布雷格曼散度框架，揭示了拟算术均值、参数族与散度泛函之间的相互作用，并将常见散度泛函与退火路径上的中间密度联系起来。