Mappings for Marginal Probabilities with Applications to Models in Statistical Physics

Mehdi Molkaraie
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Abstract

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.
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边际概率映射及其在统计物理模型中的应用
我们给出了将由其原法因子图表示的全局概率质量函数的边际概率与其对偶法因子图中相应的边际概率相关联的局部映射。该映射基于模型局部因子的傅里叶变换。给出了二维最近邻Ising模型的映射细节,证明了在二维最近邻Ising模型的相变处得到不动点的局部极值。结果进一步推广到波茨模型、时钟模型和高斯马尔可夫随机场。通过使用映射,我们可以同时将所有估计的边际概率从对偶域转换到原始域(反之亦然),如果在对偶域中可以更有效地进行边际估计,这是有利的。一个特别重要的例子是正外磁场中的铁磁伊辛模型。对于该模型,在模型的对偶法向因子图中存在一个快速混合马尔可夫链(称为子图-世界过程)来生成构型。我们的数值实验表明,所提出的程序可以在各种设置下提供更准确的全局概率质量函数的边际概率估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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