Castellana Michele, Bialek William
Joseph Henry Laboratories of Physics and Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA.
Joseph Henry Laboratories of Physics and Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA and Initiative for the Theoretical Sciences, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016, USA.
Phys Rev Lett. 2014 Sep 12;113(11):117204. doi: 10.1103/PhysRevLett.113.117204. Epub 2014 Sep 10.
If we have a system of binary variables and we measure the pairwise correlations among these variables, then the least structured or maximum entropy model for their joint distribution is an Ising model with pairwise interactions among the spins. Here we consider inhomogeneous systems in which we constrain, for example, not the full matrix of correlations, but only the distribution from which these correlations are drawn. In this sense, what we have constructed is an inverse spin glass: rather than choosing coupling constants at random from a distribution and calculating correlations, we choose the correlations from a distribution and infer the coupling constants. We argue that such models generate a block structure in the space of couplings, which provides an explicit solution of the inverse problem. This allows us to generate a phase diagram in the space of (measurable) moments of the distribution of correlations. We expect that these ideas will be most useful in building models for systems that are nonequilibrium statistical mechanics problems, such as networks of real neurons.
如果我们有一个二元变量系统，并测量这些变量之间的成对相关性，那么它们联合分布的结构最不规整或熵最大的模型就是一个自旋之间存在成对相互作用的伊辛模型。在这里，我们考虑非均匀系统，例如，在这种系统中，我们约束的不是完整的相关矩阵，而只是抽取这些相关性的分布。从这个意义上说，我们构建的是一个逆自旋玻璃：不是从一个分布中随机选择耦合常数并计算相关性，而是从一个分布中选择相关性并推断耦合常数。我们认为，这样的模型在耦合空间中生成了一个块结构，这为逆问题提供了一个明确的解决方案。这使我们能够在相关性分布的（可测量）矩空间中生成一个相图。我们预计，这些想法在构建针对诸如真实神经元网络等非平衡统计力学问题的系统模型时将最为有用。
