Maximum-entropy methods, rooted in the inverse Ising/Potts problem from statistical mechanics, have become indispensable tools for modeling pairwise interactions in disciplines such as bioinformatics, ecology, and neuroscience. Despite their remarkable success, these methods often overlook high-order interactions that may be crucial in complex systems. Conversely, while modern machine learning approaches can capture such interactions, existing interpretable frameworks are computationally expensive, making it impractical to assess the relevance of high-order interactions in real-world scenarios. Restricted Boltzmann Machines (RBMs) offer a computationally efficient alternative by encoding statistical correlations via hidden nodes in a bipartite neural network. Here, we present a method that maps RBMs exactly onto generalized Potts models with interactions of arbitrary high order. This approach leverages large-$N$ approximations, facilitated by the simple architecture of the RBM, to enable the efficient extraction of effective many-body couplings with minimal computational cost. This mapping also enables the development of a general formal framework for the extraction of effective higher-order interactions in arbitrarily complex probabilistic models. Additionally, we introduce a robust formalism for gauge fixing within the generalized Potts model. We validate our method by accurately recovering two- and three-body interactions from synthetic datasets. Additionally, applying our framework to protein sequence data demonstrates its effectiveness in reconstructing protein contact maps, achieving performance comparable to state-of-the-art inverse Potts models. These results position RBMs as a powerful and efficient tool for investigating high-order interactions in complex systems.

Generative models offer a direct way of modeling complex data. Energy-based models attempt to encode the statistical correlations observed in the data at the level of the Boltzmann weight associated with an energy function in the form of a neural network. We address here the challenge of understanding the physical interpretation of such models. In this study, we propose a simple solution by implementing a direct mapping between the Restricted Boltzmann Machine and an effective Ising spin Hamiltonian. This mapping includes interactions of all possible orders, going beyond the conventional pairwise interactions typically considered in the inverse Ising (or Boltzmann Machine) approach, and allowing the description of complex datasets. Earlier works attempted to achieve this goal, but the proposed mappings were inaccurate for inference applications, did not properly treat the complexity of the problem, or did not provide precise prescriptions for practical application. To validate our method, we performed several controlled inverse numerical experiments in which we trained the RBMs using equilibrium samples of predefined models with local external fields, 2-body and 3-body interactions in different sparse topologies. The results demonstrate the effectiveness of our proposed approach in learning the correct interaction network and pave the way for its application in modeling interesting binary variable datasets. We also evaluate the quality of the inferred model based on different training methods.

We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then cou(cid:173) pling the estimates together. The coupling model is similar to the Bradley-Terry method for paired comparisons. We study the na(cid:173) ture of the class probability estimates that arise, and examine the performance of the procedure in simulated datasets. The classifiers used include linear discriminants and nearest neighbors: applica(cid:173) tion to support vector machines is also briefly described.

Pairwise coupling is a popular multi-class classification method that combines together all pairwise comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement. We show conceptually and experimentally that the proposed approaches are more stable than two existing popular methods: voting and [3].

Multiclass probability estimation is the problem of estimating conditional probabilities of a data point belonging to a class given its covariate information. It has broad applications in statistical analysis and data science. Recently a class of weighted Support Vector Machines (wSVMs) has been developed to estimate class probabilities through ensemble learning for $K$-class problems (Wu, Zhang and Liu, 2010; Wang, Zhang and Wu, 2019), where $K$ is the number of classes. The estimators are robust and achieve high accuracy for probability estimation, but their learning is implemented through pairwise coupling, which demands polynomial time in $K$. In this paper, we propose two new learning schemes, the baseline learning and the One-vs-All (OVA) learning, to further improve wSVMs in terms of computational efficiency and estimation accuracy. In particular, the baseline learning has optimal computational complexity in the sense that it is linear in $K$. Though not being most efficient in computation, the OVA offers the best estimation accuracy among all the procedures under comparison. The resulting estimators are distribution-free and shown to be consistent. We further conduct extensive numerical experiments to demonstrate finite sample performance.

We examine several aspects of explicability of a classification system built from neural networks. The first aspect is the pairwise explicability, which is the ability to provide the most accurate prediction when the range of possibilities is narrowed to just two. Next we consider explicability in development, which means ability to make incremental improvement in prediction accuracy based on observed deficiency of the system. Inherent stochasticity of neural network based classifiers can be interpreted using likelihood randomness explicability. Finally, sureness explicability indicates confidence of the classifying system to make any prediction at all. These concepts are examined in the framework of pairwise coupling, which is a non-trainable metamodel that originated during development of support vector machines. Several methodologies are evaluated, of which the key one is shown to be the choice of the pairwise coupling method. We compare two methods: the established Wu-Lin-Weng method with the recently proposed Bayes covariant method. Our experiments indicate that the Wu-Lin-Weng method gives more weight to a single pairwise classifier, whereas the latter tries to balance information from the whole matrix of pairwise likelihoods. This translates into higher accuracy, and better sureness predictions for the Bayes covariant method. Pairwise coupling methodology has its costs, especially in terms of the number of parameters (but not necessarily in terms of training costs). However, when additional explicability aspects beyond accuracy are desired in an application, the pairwise coupling models are a promising alternative to the established methodology.

Pairwise coupling is a popular multi-class classification method that combines together all pairwise comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement. We show conceptually and experimentally that the proposed approaches are more stable than two existing popular methods: voting and [3].

Pairwise coupling is a popular multi-class classification method that combines together all pairwise comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement. We show conceptually and experimentally that the proposed approaches are more stable than two existing popular methods: voting and [3].

Pairwise coupling is a popular multi-class classification method that combines together all pairwise comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement. We show conceptually and experimentally that the proposed approaches are more stable than two existing popular methods: voting and [3].