论文标题
与统计物理模型的应用有关的边际概率的映射
Mappings for Marginal Probabilities with Applications to Models in Statistical Physics
论文作者
论文摘要
我们提出了将以其原始正常因子图表示的全局概率质量函数的边际概率与其双重正常因子图中相应的边缘概率相关的边际概率。该映射基于模型局部因素的傅立叶变换。为ISING模型提供了映射的详细信息，在该模型中证明，固定点的局部极值是在二维最近邻居ising模型的相变时实现的。结果将进一步扩展到Potts模型，时钟模型以及高斯Markov随机字段。通过采用映射，我们可以同时将所有估计的边际概率从双重域转换为原始域（反之亦然），如果可以在双重域中更有效地进行边缘，这是有利的。具有特殊意义的一个例子是在正外磁场中的铁磁ising模型。对于此模型，存在一个快速混合的马尔可夫链（称为子图 - 世界过程），以在模型的双重正常因子图中生成配置。我们的数值实验表明，所提出的程序可以提供更准确的估计，以估计各种设置中全局概率质量函数的边际概率。
We present local mappings that relate the marginal probabilities of a global probability mass function represented by its primal normal factor graph to the corresponding marginal probabilities in its dual normal factor graph. The mapping is based on the Fourier transform of the local factors of the models. Details of the mapping are provided for the Ising model, where it is proved that the local extrema of the fixed points are attained at the phase transition of the two-dimensional nearest-neighbor Ising model. The results are further extended to the Potts model, to the clock model, and to Gaussian Markov random fields. By employing the mapping, we can transform simultaneously all the estimated marginal probabilities from the dual domain to the primal domain (and vice versa), which is advantageous if estimating the marginals can be carried out more efficiently in the dual domain. An example of particular significance is the ferromagnetic Ising model in a positive external magnetic field. For this model, there exists a rapidly mixing Markov chain (called the subgraphs--world process) to generate configurations in the dual normal factor graph of the model. Our numerical experiments illustrate that the proposed procedure can provide more accurate estimates of marginal probabilities of a global probability mass function in various settings.
