In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain.

翻译：本文评估了马尔可夫基方法在条件分布采样中的挑战与最佳实践。在Diaconis与Sturmfels发表马尔可夫基本定理25周年之际，我们提供了相关见解与澄清。除文献综述外，我们还在层级模型中马尔可夫基的复杂性、对数线性模型纤维的松弛以及部分移动集在提供不可约马尔可夫链中的局限性等方面，证明了三个新的结论。