Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

翻译：高维数据建模充满挑战，但却是理解许多复杂系统的关键。最大熵模型（如伊辛模型和Potts模型）已被广泛用于从数据中的相关模式捕获成对相互作用，从而能够从观测值（例如蛋白质序列或神经群体活动）中推断复杂系统的图形表示。近年来，人们对同时涉及三个或更多变量的高阶相关模式建模的兴趣日益增长。虽然已在高阶伊辛模型的二元数据处理方面取得进展，但我们将此框架扩展到更一般的离散数据情形。我们引入了q态自旋模型——一个完整的最大熵模型族，它通过包含远程和任意高阶相互作用来推广向量Potts模型。在成对情形下，与标准向量Potts模型相比，我们的模型允许更多样化的相互作用类型。我们通过实例讨论了其统计解释，并将其与离散傅里叶分析联系起来。利用配分函数的圈展开，我们证明了自旋模型的统计性质完全由其相互作用的代数结构决定。我们定义了规范变换，在该变换下这一结构以及配分函数保持不变。规范等价的模型可被视为同一抽象统计模型的不同表示，尽管它们通常具有不同阶的相互作用，这扩展了二元情形下的结论。为便于实际数据分析应用，我们聚焦于二元情形下称为"最小复杂模型"的子集，并将其推广到离散数据。我们获得了这些模型边际似然的闭式表达式，从而能够快速进行模型选择。我们通过简单的现实案例展示了其应用。