Multivariate Bernoulli Hoeffding Decomposition: From Theory to Sensitivity Analysis

• URL: http://arxiv.org/abs/2510.07088v3
• Date: Wed, 05 Nov 2025 12:47:58 GMT
• Title: Multivariate Bernoulli Hoeffding Decomposition: From Theory to Sensitivity Analysis
• Authors: Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa, Jean-Michel Loubes, Joseph Muré,
• Abstract summary: This work focuses on the case of Bernoulli inputs and provides a complete analytical characterization of the decomposition.<n>We show that, in this discrete setting, the associated subspaces are one-dimensional and that the decomposition admits a closed-form representation.<n>The paper concludes with perspectives on extending the methodology to high-dimensional settings and to models involving inputs with finite, non-binary support.
• Score: 2.762021507766656
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding the behavior of predictive models with random inputs can be achieved through functional decompositions into sub-models that capture interpretable effects of input groups. Building on recent advances in uncertainty quantification, the existence and uniqueness of a generalized Hoeffding decomposition have been established for correlated input variables, using oblique projections onto suitable functional subspaces. This work focuses on the case of Bernoulli inputs and provides a complete analytical characterization of the decomposition. We show that, in this discrete setting, the associated subspaces are one-dimensional and that the decomposition admits a closed-form representation. One of the main contributions of this study is to generalize the classical Fourier--Walsh--Hadamard decomposition for pseudo-Boolean functions to the correlated case, yielding an oblique version when the underlying distribution is not a product measure, and recovering the standard orthogonal form when independence holds. This explicit structure offers a fully interpretable framework, clarifying the contribution of each input combination and theoretically enabling model reverse engineering. From this formulation, explicit sensitivity measures-such as Sobol' indices and Shapley effects-can be directly derived. Numerical experiments illustrate the practical interest of the approach for decision-support problems involving binary features. The paper concludes with perspectives on extending the methodology to high-dimensional settings and to models involving inputs with finite, non-binary support.

Related papers

• Exact Functional ANOVA Decomposition for Categorical Inputs Models [2.762021507766656]
  ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions.<n>For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability.<n>We completely resolve this limitation for categorical inputs.
  arXiv  Detail & Related papers  (2026-03-03T06:59:56Z)
• Loss-Complexity Landscape and Model Structure Functions [53.92822954974537]
  We develop a framework for dualizing the Kolmogorov structure function $h_x(alpha)$.<n>We establish a mathematical analogy between information-theoretic constructs and statistical mechanics.<n>We explicitly prove the Legendre-Fenchel duality between the structure function and free energy.
  arXiv  Detail & Related papers  (2025-07-17T21:31:45Z)
• Emergence of Quantised Representations Isolated to Anisotropic Functions [0.0]
  This paper presents a novel methodology for determining representational structure, which builds upon the existing Spotlight Resonance method.<n>It shows how discrete representations can emerge and organise in autoencoder models, through a controlled ablation study in which only the activation function is altered.<n>Using this technique, the validity of whether function-driven symmetries can act as implicit inductive biases on representations is determined.
  arXiv  Detail & Related papers  (2025-07-16T09:27:54Z)
• Effect Decomposition of Functional-Output Computer Experiments via Orthogonal Additive Gaussian Processes [8.723426955657347]
  Functional ANOVA (FANOVA) is a widely used variance-based sensitivity analysis tool.<n>This study proposes a functional-output orthogonal additive Gaussian process (FOAGP) to efficiently perform the data-driven orthogonal effect decomposition.<n>The FOAGP framework also provides analytical formulations for local Sobol' indices and expected conditional variance sensitivity indices.
  arXiv  Detail & Related papers  (2025-06-15T03:24:55Z)
• Internal Causal Mechanisms Robustly Predict Language Model Out-of-Distribution Behaviors [61.92704516732144]
  We show that the most robust features for correctness prediction are those that play a distinctive causal role in the model's behavior.<n>We propose two methods that leverage causal mechanisms to predict the correctness of model outputs.
  arXiv  Detail & Related papers  (2025-05-17T00:31:39Z)
• I Predict Therefore I Am: Is Next Token Prediction Enough to Learn Human-Interpretable Concepts from Data? [76.15163242945813]
  Large language models (LLMs) have led many to conclude that they exhibit a form of intelligence.<n>We introduce a novel generative model that generates tokens on the basis of human-interpretable concepts represented as latent discrete variables.
  arXiv  Detail & Related papers  (2025-03-12T01:21:17Z)
• Bilinear Convolution Decomposition for Causal RL Interpretability [0.0]
  Efforts to interpret reinforcement learning (RL) models often rely on high-level techniques such as attribution or probing.<n>This work proposes replacing nonlinearities in convolutional neural networks (ConvNets) with bilinear variants, to produce a class of models for which these limitations can be addressed.<n>We show bilinear model variants perform comparably in model-free reinforcement learning settings, and give a side by side comparison on ProcGen environments.
  arXiv  Detail & Related papers  (2024-12-01T19:32:04Z)
• Unsupervised Representation Learning from Sparse Transformation Analysis [79.94858534887801]
  We propose to learn representations from sequence data by factorizing the transformations of the latent variables into sparse components. Input data are first encoded as distributions of latent activations and subsequently transformed using a probability flow model.
  arXiv  Detail & Related papers  (2024-10-07T23:53:25Z)
• Scaling and renormalization in high-dimensional regression [72.59731158970894]
  We present a unifying perspective on recent results on ridge regression.<n>We use the basic tools of random matrix theory and free probability, aimed at readers with backgrounds in physics and deep learning.<n>Our results extend and provide a unifying perspective on earlier models of scaling laws.
  arXiv  Detail & Related papers  (2024-05-01T15:59:00Z)
• Identifiable Latent Neural Causal Models [82.14087963690561]
  Causal representation learning seeks to uncover latent, high-level causal representations from low-level observed data. We determine the types of distribution shifts that do contribute to the identifiability of causal representations. We translate our findings into a practical algorithm, allowing for the acquisition of reliable latent causal representations.
  arXiv  Detail & Related papers  (2024-03-23T04:13:55Z)
• Understanding Diffusion Models by Feynman's Path Integral [2.4373900721120285]
  We introduce a novel formulation of diffusion models using Feynman's integral path. We find this formulation providing comprehensive descriptions of score-based generative models. We also demonstrate the derivation of backward differential equations and loss functions.
  arXiv  Detail & Related papers  (2024-03-17T16:24:29Z)
• Hoeffding decomposition of black-box models with dependent inputs [30.076357972854723]
  We generalize Hoeffding's decomposition for dependent inputs under mild conditions. We show that any square-integrable, real-valued function of random elements respecting two assumptions can be uniquely additively and offer a characterization.
  arXiv  Detail & Related papers  (2023-10-10T12:28:53Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling.<n>This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.<n>We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Measuring dissimilarity with diffeomorphism invariance [94.02751799024684]
  We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces. We prove that DID enjoys properties which make it relevant for theoretical study and practical use.
  arXiv  Detail & Related papers  (2022-02-11T13:51:30Z)
• An Indirect Rate-Distortion Characterization for Semantic Sources: General Model and the Case of Gaussian Observation [83.93224401261068]
  Source model is motivated by the recent surge of interest in the semantic aspect of information. intrinsic state corresponds to the semantic feature of the source, which in general is not observable. Rate-distortion function is the semantic rate-distortion function of the source.
  arXiv  Detail & Related papers  (2022-01-29T02:14:24Z)
• Understanding Neural Abstractive Summarization Models via Uncertainty [54.37665950633147]
  seq2seq abstractive summarization models generate text in a free-form manner. We study the entropy, or uncertainty, of the model's token-level predictions. We show that uncertainty is a useful perspective for analyzing summarization and text generation models more broadly.
  arXiv  Detail & Related papers  (2020-10-15T16:57:27Z)
• Exponentially Weighted l_2 Regularization Strategy in Constructing Reinforced Second-order Fuzzy Rule-based Model [72.57056258027336]
  In the conventional Takagi-Sugeno-Kang (TSK)-type fuzzy models, constant or linear functions are usually utilized as the consequent parts of the fuzzy rules. We introduce an exponential weight approach inspired by the weight function theory encountered in harmonic analysis.
  arXiv  Detail & Related papers  (2020-07-02T15:42:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.