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 work has constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. We provide a comprehensive analysis of this 'centroid' property using Bregman divergences under a monotonic embedding of the density function, thereby associating common divergences such as Amari's and Renyi's ${\alpha}$-divergences, ${(\alpha,\beta)}$-divergences, and the Jensen-Shannon divergence with intermediate densities along an annealing path. Our analysis highlights the interplay between parametric families, quasi-arithmetic means, and divergence functions using the rho-tau Bregman divergence framework of Zhang 2004,2013.

翻译：从复杂分布中采样并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目标密度之间建立桥梁。已有研究利用准算术均值构建退火路径，并将所得中间密度解释为最小化与端点间的期望散度。我们通过密度函数的单调嵌入，运用Bregman散度对这一"质心"性质展开全面分析，从而将常见的散度（如Amari散度和Rényi的${\alpha}$-散度、${(\alpha,\beta)}$-散度以及Jensen-Shannon散度）与退火路径上的中间密度相关联。本研究利用Zhang（2004, 2013）提出的rho-tau Bregman散度框架，揭示了参数族、准算术均值与散度函数之间的内在关联。