Susceptibility Propagation by Using Diagonal Consistency

There is an increased demand for techniques that can be used to evaluate local statistical quantities such as local magnetizations and local susceptibilities(covariances) of spin systems with finite sizes in statistical physics as well as in other scientific fields. This is because the use and application of Markov random fields and probabilistic operations is widely prevalent in many areas, particularly in the field of information sciences [1, 2]. Linear response methods have been used to approximately evaluate pair correlations between non-nearest pairs by combining an advanced mean-field method, particularly the cluster variation method, with the linear response theory [3, 4]. In the current decade, a suitable technique called susceptibility propagation (SusP) (also called a variational linear response in the field of information sciences), has been developed to compute the local susceptibilities of Markov random fields [5-8]. In general, SusPs are constructed by combining belief propagation algorithms and linear response methods. Belief propagation algorithms are one of the most popular message-passing type of algorithms that is used to approximately compute local magnetizations of Markov random fields [9], and they are equivalent to Bethe approximations [10] used in statistical physics [11, 12]. It is known that SusPs can be used to compute local susceptibilities with a high degree of accuracy because linear response methods partially restore the effects of loops of networks that are neglected in the Bethe approximations[13]. SusPs have the following inconsistency due to approximation.